Introduction
We consider fair allocations of m m m n n n M M M m m m N : = [ n ] N := [n] N := [ n ] i ∈ N i \in N i ∈ N valuation function v i v_i v i M M M v i v_i v i v i ( ∅ ) = 0 v_i(\varnothing)=0 v i ( ∅ ) = 0 S 1 ⊆ S 2 S_1 \subseteq S_2 S 1 ⊆ S 2 v i ( S 1 ) ≤ v i ( S 2 ) v_i(S_1) \le v_i(S_2) v i ( S 1 ) ≤ v i ( S 2 ) v i v_i v i v i v_i v i additive : for every S ⊆ M S \subseteq M S ⊆ M v i ( S ) = ∑ e ∈ S v i ( e ) v_i(S)=\sum_{e \in S}v_i(e) v i ( S ) = ∑ e ∈ S v i ( e )
An allocation of items M M M N N N ( P 1 , … , P n ) (P_1,\ldots,P_n) ( P 1 , … , P n ) M M M n n n i i i P i P_i P i fair , for a certain definition of fairness. In this work, we will focus on a share-based notion of fairness called the maximin share [9 ] , or MMS for short.
Definition 1.1. Let P n \mathcal{P}_n P n ( P 1 , … , P n ) (P_1,\ldots,P_n) ( P 1 , … , P n ) M M M n n n i i i v i v_i v i maximin share is
MMS ( M , v i , n ) = max ( P 1 , … , P n ) ∈ P n min j ∈ [ n ] v i ( P j ) . \operatorname{MMS}(M,v_i,n)=\max_{(P_1,\ldots,P_n)\in\mathcal{P}_n}\min_{j\in[n]}v_i(P_j). MMS ( M , v i , n ) = ( P 1 , … , P n ) ∈ P n max j ∈ [ n ] min v i ( P j ) . We will use MMS ( n ) \operatorname{MMS}(n) MMS ( n ) MMS-value with respect to partitions of M M M n n n
A partition ( A 1 , … , A n ) (A_1,\ldots,A_n) ( A 1 , … , A n ) M M M MMS-allocation if for every agent i ∈ N i \in N i ∈ N v i ( A i ) ≥ MMS ( M , v i , n ) v_i(A_i) \ge\operatorname{MMS}(M,v_i,n) v i ( A i ) ≥ MMS ( M , v i , n ) i i i A i A_i A i ρ ∈ ( 0 , 1 ] \rho\in(0,1] ρ ∈ ( 0 , 1 ] ( A 1 , … , A n ) (A_1,\ldots,A_n) ( A 1 , … , A n ) ρ \rho ρ i ∈ N i \in N i ∈ N v i ( A i ) ≥ ρ ⋅ MMS ( M , v i , n ) v_i(A_i) \ge\rho\cdot\operatorname{MMS}(M,v_i,n) v i ( A i ) ≥ ρ ⋅ MMS ( M , v i , n )
Our results, put in context
Our results, put in context
There are known examples showing that with additive valuations, MMS allocations need not exist. The first such examples were given in [20 ] . Later examples gave explicit upper bounds on ρ \rho ρ ρ \rho ρ ρ ≤ 39 40 \rho\le\frac{39}{40} ρ ≤ 40 39 n = 3 n = 3 n = 3 [16 ] , and ρ ≤ 20 21 \rho\le\frac{20}{21} ρ ≤ 21 20 n = 4 n = 4 n = 4 [13 ] . However, as n n n ρ \rho ρ
Conjecture 1. For every ρ < 1 \rho< 1 ρ < 1 n ρ n_\rho n ρ n ≥ n ρ n \ge n_\rho n ≥ n ρ ρ \rho ρ
We currently do not have strong beliefs as to whether the conjecture is true. So far, attempts to refute the conjecture have not been successful. However, they did provide examples of allocation instances with successively stronger lower bounds on the possible value of n ρ n_\rho n ρ ρ \rho ρ ρ \rho ρ 1 − ρ 1-\rho 1 − ρ n n n [20 ] the slackness was exponentially small in 1 / n 1/n 1/ n [16 ] it was improved to be polynomially small, and in [13 ] it was further improved to be polylogarithmically small. To refute the conjecture, the slackness needs to be a constant independent of n n n
A recurring feature in all the above negative examples is that they allow what we refer to as agent redundancy . That is, for each of them, for every n n n ρ n \rho_n ρ n d > n d > n d > n d d d n n n n n n ρ n \rho_n ρ n ( n ) (n) ( n ) [20 ] and [16 ] , d d d 2 n 2n 2 n [13 ] d d d n n n n n n ρ n \rho_n ρ n
Our main positive result proves Conjecture 1, but only in the setting with agent redundancy.
Theorem 1.1. For every ε > 0 \varepsilon> 0 ε > 0 ρ < 1 \rho< 1 ρ < 1 n ε , ρ n_{\varepsilon,\rho} n ε , ρ n ≥ n ε , ρ n \ge n_{\varepsilon,\rho} n ≥ n ε , ρ d ≥ ( 1 + ε ) n d \ge(1+\varepsilon)n d ≥ ( 1 + ε ) n n n n ρ \rho ρ ( n ) (n) ( n )
Note that in Theorem 1.1, every agent i i i ρ ⋅ MMS ( M , v i , n ) \rho\cdot\operatorname{MMS}(M,v_i,n) ρ ⋅ MMS ( M , v i , n ) ρ ⋅ MMS ( M , v i , d ) \rho\cdot\operatorname{MMS}(M,v_i,d) ρ ⋅ MMS ( M , v i , d ) d ≥ n d \ge n d ≥ n MMS ( M , v i , n ) ≥ MMS ( M , v i , d ) \operatorname{MMS}(M,v_i,n) \ge\operatorname{MMS}(M,v_i,d) MMS ( M , v i , n ) ≥ MMS ( M , v i , d ) d > n d > n d > n d = n + 1 d = n+1 d = n + 1
A straightforward corollary of Theorem 1.1 is that Conjecture 1 holds when the number of items m m m 2 n 2n 2 n
Corollary 1.1. For every ε > 0 \varepsilon> 0 ε > 0 ρ < 1 \rho< 1 ρ < 1 n ε , ρ n_{\varepsilon,\rho} n ε , ρ n ≥ n ε , ρ n \ge n_{\varepsilon,\rho} n ≥ n ε , ρ m ≤ ( 2 − ε ) n m \le(2-\varepsilon)n m ≤ ( 2 − ε ) n ρ \rho ρ
We note that for ρ = 1 \rho= 1 ρ = 1 m ≤ n + log n m \le n+\log n m ≤ n + log n [14 ] , and known not to hold when m ≥ n + c ⋅ log n m \ge n+c\cdot\log n m ≥ n + c ⋅ log n c c c [13 ] .
A closely related topic is a so called 1-out-of-d d d d > n d>n d > n ordinal MMS , first introduced in [7 ] (though also considered earlier in [9 ] with d = n + 1 d=n+1 d = n + 1 ( A 1 , … , A n ) (A_1,\ldots,A_n) ( A 1 , … , A n ) M M M d d d i ∈ N i \in N i ∈ N v i ( A i ) ≥ MMS ( M , v i , d ) v_i(A_i) \ge\operatorname{MMS}(M,v_i,d) v i ( A i ) ≥ MMS ( M , v i , d ) d d d n n n ( d ) (d) ( d ) d = 4 ⌈ n / 3 ⌉ d = 4\lceil n/3\rceil d = 4 ⌈ n /3 ⌉ [12 ] showed that only sublinear number of additional bundles is needed, with d = n + O ( n log n ) d=n+O(\sqrt{n\log n}) d = n + O ( n log n )
We further reduce the required number of additional bundles to only polylogarithmic in n n n
Theorem 1.2. Any instance with n n n ( d ) (d) ( d ) d = n + O ( log 2 n ) d = n+O(\log^2 n) d = n + O ( log 2 n )
Our Theorem 1.2 proof uses essentially the same techniques as our proof for Theorem 1.1. We note that any results on 1-out-of-d d d [12 ] or Theorem 1.2) have the following interpretation in the setting with agent redundancy, discussed earlier.
Corollary 1.2. For every ε > 0 \varepsilon> 0 ε > 0 n ε n_{\varepsilon} n ε n ≥ n ε n \ge n_{\varepsilon} n ≥ n ε d ≥ ( 1 + ε ) n d \ge(1+\varepsilon)n d ≥ ( 1 + ε ) n n n n d d d
In some aspects, Corollary 1.2 is stronger than Theorem 1.1: it applies to every subset of size n n n n n n ρ < 1 \rho< 1 ρ < 1 n n n d d d
Overview of our algorithm
Here we explain the main steps of our algorithm that proves Theorem 1.1, the proof of Theorem 1.2 is easier and requires only a few of the ideas of Theorem 1.1 (even though later we will first give a full proof of Theorem 1.2 and only then Theorem 1.1). In the description, we slightly change notation. The total number of agents is n n n d d d n n n d d d d = ( 1 − ε ) n d=(1-\varepsilon)n d = ( 1 − ε ) n q q q ρ ≤ 1 − 2 q \rho\le1-\frac{2}{q} ρ ≤ 1 − q 2 ( 1 − 2 q ) (1-\frac{2}{q}) ( 1 − q 2 ) d d d
We make the following assumptions, standard in designing approximate MMS allocations.
Assumption 1. Without loss of generality, in the setting of Theorem 1.1 we can assume that
for every agent i ∈ N i \in N i ∈ N e ∈ M e \in M e ∈ M v i ( e ) ∈ [ 0 , 1 ] v_i(e) \in[0,1] v i ( e ) ∈ [ 0 , 1 ]
for every i ∈ N i \in N i ∈ N d d d B i : = ( B i , 1 , … , B i , d ) \mathcal{B}_i := (B_{i,1},\ldots,B_{i,d}) B i := ( B i , 1 , … , B i , d ) M M M M , v i , d M,v_i,d M , v i , d
∀ j ∈ [ d ] : v i ( B i , j ) = 1 ; \forall j \in[d] : v_i(B_{i,j}) = 1; ∀ j ∈ [ d ] : v i ( B i , j ) = 1 ; We refer to the bundles in this d d d i i i
([BL16]) items M = { e 1 , … , e m } M=\{e_1,\ldots,e_m\} M = { e 1 , … , e m } i ∈ N i \in N i ∈ N
v i ( e 1 ) ≥ v i ( e 2 ) ≥ … ≥ v i ( e m ) . v_i(e_1) \ge v_i(e_2) \ge\ldots\ge v_i(e_m). v i ( e 1 ) ≥ v i ( e 2 ) ≥ … ≥ v i ( e m ) . Under Assumption 1 , every selected agent should be given a value of at least 1 − 2 / q 1-2/q 1 − 2/ q
Head and Tail items. For every agent i i i head items { e 1 , … , e h i } \{e_1,\ldots,e_{h_i}\} { e 1 , … , e h i } 1 / q 1/q 1/ q tail items { e h i + 1 , … , m } \{e_{h_i+1},\ldots,m\} { e h i + 1 , … , m } 1 / q 1/q 1/ q 2 / q 2/q 2/ q 1 / q 1/q 1/ q 1 / q 2 1/q^2 1/ q 2
Only tail items. Consider first the case that all items are tail items for all agents. Namely, every item has value at most 1 / q 1/q 1/ q d d d { 1 , … , d } ∈ N \{1,\ldots,d\} \in N { 1 , … , d } ∈ N i i i e j e_j e j j = k ⋅ d + i j=k\cdot d+i j = k ⋅ d + i k = { 0 , 1 , 2 , … } k=\{0,1,2,\ldots\} k = { 0 , 1 , 2 , … } i i i v i ( M ) / d − v i ( e 1 ) ≥ 1 − 1 / q v_i(M)/d-v_i(e_1) \ge1-1/q v i ( M ) / d − v i ( e 1 ) ≥ 1 − 1/ q Assumption 1 ).
Only head items. Consider now the case that all items are head items. Namely, every item has value at least 1 / q 1/q 1/ q
At a very high level, our plan has two steps. In step one, for some small ε ′ \varepsilon' ε ′ ( 1 − ε ′ ) d (1-\varepsilon')d ( 1 − ε ′ ) d d d d ε ′ n \varepsilon'n ε ′ n ( ε ′ d ) (\varepsilon'd) ( ε ′ d )
The main difficulty in our proof is to show that the second step can be done successfully. The fact that the first step by itself can be done is an immediate consequence of earlier work, specifically the variation of Corollary 1.2 by [12 ] . However, due to the need to enable the second step, we do not use Corollary 1.2, and instead do the first step differently. Actually, already in that step we use the fact that we may give selected agents only ρ \rho ρ
Profiles and breakpoints. Round down the value of each item to the nearest multiple of 1 / q 2 1/q^2 1/ q 2 q 2 q^2 q 2 { 1 q , 1 q + 1 q 2 , … , 1 } \left\{\frac{1}{q},\frac{1}{q}+\frac{1}{q^2},\ldots,1\right\} { q 1 , q 1 + q 2 1 , … , 1 } q q q 1 − q ⋅ 1 q 2 = 1 − 1 / q 1-q\cdot\frac{1}{q^2}=1-1/q 1 − q ⋅ q 2 1 = 1 − 1/ q
With each MMS bundle we associate a profile , which replaces each item with a corresponding value of its class. As there are at most q q q q 2 q^2 q 2 q 2 q q^{2q} q 2 q n n n n n n
Recall that items are ordered in order of decreasing value. For every agent i i i e j e_j e j breakpoint b i j b_{ij} b ij ℓ \ell ℓ v i ( e ℓ ) ≥ v i ( e j ) v_i(e_\ell)\ge v_i(e_j) v i ( e ℓ ) ≥ v i ( e j ) ℓ ≥ j \ell\ge j ℓ ≥ j ℓ > j \ell>j ℓ > j q 2 q^2 q 2 q 2 q^2 q 2
For each agent, for each class of items there is a single breakpoint location associated with all its items. With each profile of item classes, we can associate a profile of respective breakpoint locations, where the multiplicity of a breakpoint equals the multiplicity of the respective class in the profile.
Given t ≤ n t\le n t ≤ n 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m ℓ \ell ℓ load on ℓ \ell ℓ ℓ \ell ℓ
Importantly, even if the collection of selected MMS bundles was not feasible (that is, some items appear in more than one selected bundle), it may still happen that the collection of breakpoint profiles is feasible. This is because we need not give each agent the exact items appearing in her MMS bundle. We can replace each item e j e_j e j e ℓ e_\ell e ℓ v i ( e ℓ ) < v i ( e j ) v_i(e_\ell)<v_i(e_j) v i ( e ℓ ) < v i ( e j )
We summarize the above discussion in the following lemma.
Lemma 1.1. Let A A A 0 ≤ ℓ ≤ m 0 \le\ell\le m 0 ≤ ℓ ≤ m ℓ \ell ℓ ℓ \ell ℓ A A A ( 1 − 1 q ) (1-\frac{1}{q}) ( 1 − q 1 )
Satisfying nearly d d d Before describing how we actually do the first step of our algorithm (for the case that there are only head items), it is instructive to see how one can do that step if there is no need to enable the future second step of our algorithm. The proof in our setting is considerably simpler than the proof for a variation of Corollary 1.2 given in [12 ] , and gives a much smaller value of ε \varepsilon ε
Take d d d d − O ( q 2 log d ) d-O(q^2\log d) d − O ( q 2 log d ) O ( log d ) O(\log d) O ( log d ) O ( q 2 ) O(q^2) O ( q 2 ) log d \log d log d O ( q 2 log d ) O(q^2\log d) O ( q 2 log d ) O ( q 2 log d ) O(q^2\log d) O ( q 2 log d ) d − O ( q 2 log d ) d-O(q^2\log d) d − O ( q 2 log d ) § 10 , instead of discarding all the agents at the end, we discard O ( q 2 ) O(q^2) O ( q 2 )
In the first iteration, we start with the fractional allocation in which each agent selects a 1 q \frac{1}{q} q 1 d d d d 2 d^2 d 2
More generally, also at later iterations, we will have some d ′ d' d ′ f ′ ≥ 2 d ′ f' \ge2d' f ′ ≥ 2 d ′ q q q q f ′ qf' q f ′
Now, instead of considering all load constraints, select f ′ − d ′ 2 \frac{f'-d'}{2} 2 f ′ − d ′ checkpoints , and consider the loads only at these checkpoints. The checkpoints are selected so that they are “evenly spaced”, in the sense that between any two consecutive checkpoints, at most O ( q ) O(q) O ( q ) d ′ d' d ′ f ′ + d ′ 2 \frac{f'+d'}{2} 2 f ′ + d ′ f ′ − d ′ 2 \frac{f'-d'}{2} 2 f ′ − d ′ f ′ + d ′ 2 ≤ 3 f ′ 4 \frac{f'+d'}{2} \le\frac{3f'}{4} 2 f ′ + d ′ ≤ 4 3 f ′ f ′ ≥ 2 d ′ f' \ge2d' f ′ ≥ 2 d ′ O ( q 2 ) O(q^2) O ( q 2 ) O ( q ) O(q) O ( q ) q q q § 10 .
As a consequence, we have the following proposition.
Proposition 1.1. For every positive integer q q q n n n 1 q \frac{1}{q} q 1 ( 1 − 1 q ) (1-\frac{1}{q}) ( 1 − q 1 ) n − O ( q 2 log n ) n-O(q^2\log n) n − O ( q 2 log n )
The first step, satisfying a set A A A ( 1 − ε ′ ) d (1-\varepsilon')d ( 1 − ε ′ ) d In this step, for some small ε ′ \varepsilon' ε ′ A A A ( 1 − ε ′ ) d (1-\varepsilon')d ( 1 − ε ′ ) d
We will partition the set of agents into three sets: a large set V V V ( 1 − ε 4 ) n (1-\frac{\varepsilon}{4})n ( 1 − 4 ε ) n W W W ε n 8 \frac{\varepsilon n}{8} 8 ε n U U U ε n 8 \frac{\varepsilon n}{8} 8 ε n A A A V V V
In addition, we partition the set of items into two sets, E E E S S S S S S n n n f ( q , ε ) log n f(q,\varepsilon)\log n f ( q , ε ) log n f f f q q q ε \varepsilon ε n n n E E E A A A S S S
Considering the sorted order of items in M M M S S S O ( q 2 ) O(q^2) O ( q 2 ) e 1 e_1 e 1 f ( q , ε ) log n f(q,\varepsilon)\log n f ( q , ε ) log n S S S ∣ S ∣ |S| ∣ S ∣ S ′ S' S ′ E E E U ∪ V U\cup V U ∪ V W W W ∣ W ∣ ≤ ε n 8 |W|\leq\frac{\varepsilon n}{8} ∣ W ∣ ≤ 8 ε n S S S S ′ S' S ′ S ′ S' S ′ S S S n 2 \frac{n}{2} 2 n § 5.2 for more details.
We sketch how the set S S S q 2 q^2 q 2 q 2 n q^2n q 2 n 2 q 2 2q^2 2 q 2 e 1 e_1 e 1 n 2 \frac{n}{2} 2 n S S S 2 q 2 2q^2 2 q 2 S S S Θ ( q 4 ) log n \Theta(q^4)\log n Θ ( q 4 ) log n q O ( q 2 ) q^{O(q^2)} q O ( q 2 ) ε n 8 \frac{\varepsilon n}{8} 8 ε n 8 q 2 ε \frac{8q^2}{\varepsilon} ε 8 q 2 S ′ S' S ′ S S S ∣ S ∣ |S| ∣ S ∣ q O ( q 4 / ε ) log n q^{O(q^4/\varepsilon)}\log n q O ( q 4 / ε ) log n n n n q q q S S S n 2 \frac{n}{2} 2 n A A A S S S
Now, we explain why agents of V V V E = M − S E=M-S E = M − S n n n ∣ S ∣ |S| ∣ S ∣ n n n d d d S S S n ≥ d ( 1 + ε ) n\geq d(1+\varepsilon) n ≥ d ( 1 + ε ) ε d \varepsilon d ε d d d d S S S
At the end of the first step, we will have an allocation of items from E E E A A A
Agents from A A A S ′ S' S ′ ℓ \ell ℓ E − S ′ E-S' E − S ′ A A A E E E ℓ \ell ℓ
To get some intuition why we would have a large slackness property above, consider a uniform fractional allocation over the MMS partitions. Observe that at most q 2 q^2 q 2 ℓ \ell ℓ 0 0 0 ℓ \ell ℓ q 2 n q^2n q 2 n ℓ ∈ [ m ] \ell\in[m] ℓ ∈ [ m ] 1 n ⋅ m 2 q 2 \frac{1}{n} \cdot\frac{m}{2q^2} n 1 ⋅ 2 q 2 m [ m ] [m] [ m ] q 2 q^2 q 2 t t t ( t 2 ) \binom{t}{2} ( 2 t ) m 2 2 q 2 \frac{m^2}{2q^2} 2 q 2 m 2 ℓ ∈ [ m ] \ell\in[m] ℓ ∈ [ m ] 1 n ⋅ m 2 q 2 \frac{1}{n} \cdot\frac{m}{2q^2} n 1 ⋅ 2 q 2 m s s s s s s 2 s q 2 2sq^2 2 s q 2 A A A
The second step, adding ε ′ d \varepsilon'd ε ′ d Recall the sets of agents U , V , W U,V,W U , V , W E , S E,S E , S E E E A ⊂ V A \subset V A ⊂ V ∣ A ∣ = ( 1 − ε ′ ) d |A|=(1-\varepsilon')d ∣ A ∣ = ( 1 − ε ′ ) d ε ′ d ≤ O ( q 2 log n ) \varepsilon'd \le O(q^2\log n) ε ′ d ≤ O ( q 2 log n ) ε ′ d \varepsilon'd ε ′ d A A A U U U
Remark 1.1. As we also have agents remaining in V − A V-A V − A U U U U U U V − A V-A V − A U U U ε n 8 \frac{\varepsilon n}{8} 8 ε n
We partition the agents in U U U S S S S S S ∣ S ∣ |S| ∣ S ∣ S S S 2 q 2 2q^2 2 q 2 ∣ S ∣ > 2 q 2 |S|>2q^2 ∣ S ∣ > 2 q 2 ∣ S ∣ |S| ∣ S ∣
The class of an agent is determined by the locations of its breakpoints. Each breakpoint can be in one of the classes of items, and thus we have at most ∣ S ∣ q 2 |S|^{q^2} ∣ S ∣ q 2 q q q ε \varepsilon ε n n n ∣ S ∣ ≤ q O ( q 4 / ε ) log n |S|\le q^{O(q^4/\varepsilon)}\log n ∣ S ∣ ≤ q O ( q 4 / ε ) log n n n n U U U O ( q 2 log n ) O(q^2\log n) O ( q 2 log n ) A A A U ′ U' U ′ U U U U U U
Our way of analyzing the existence of allocations is based on locations of breakpoints and loads at selected checkpoints. The checkpoints that interest us in the second step are those at the beginning of each S ′ S' S ′ U ′ U' U ′ U ′ U' U ′ U ′ U' U ′ S S S U U U
It will be convenient for us to take an arbitrary agent r ∈ U ′ r \in U' r ∈ U ′ r r r d d d 0 ≤ t ≤ ∣ A ∣ 0 \le t \le|A| 0 ≤ t ≤ ∣ A ∣ A − A^- A − t t t A A A B + B^+ B + t + 1 t+1 t + 1 r r r A A A A − A^- A − B + B^+ B + t = ∣ A ∣ t=|A| t = ∣ A ∣ A A A ∣ A ∣ + 1 |A|+1 ∣ A ∣ + 1 r r r t t t t < ∣ U ′ ∣ t<|U'| t < ∣ U ′ ∣
The answer to this question in our setting is yes, because the number of checkpoints depends only on q q q [11 ] (though apparently also proved by others). That lemma states that any collection of non-negative integer vectors of finite dimension has a finite set of minimal vectors (that do not dominate other vectors). We represent feasible solutions to our problem as integer vectors of dimension that depends only on q q q q q q t t t q q q ∣ U ′ ∣ |U'| ∣ U ′ ∣ n n n
The solution with t < ∣ U ′ ∣ t < |U'| t < ∣ U ′ ∣ B + B^+ B + r r r t + 1 t+1 t + 1 U ′ U' U ′ j j j j j j j j j B + B^+ B + U ′ U' U ′
We remark that MMS bundles of agents from U ′ U' U ′ S S S S S S S ′ S' S ′ A A A
Mixture of head and tail items. We have seen (an overview of) how to prove Theorem 1.1 when either all items are head items, or all items are tail items. We now explain how to handle instances with combination of head items and tail items. We basically follow the approach outlined for the case of only head items, but with some adjustments. Note that the same item can be a head item for some agents, but a tail item for other agents, a fact that somewhat complicates the situation.
We shall use the known fact that fractional allocations can be rounded to integer allocations, where each agent receives a value that differs from that of her fractional allocation by the value of at most one item. Hence, for tail items (for which a single item has value at most 1 q \frac{1}{q} q 1
For each agent i i i T i T_i T i
For each bundle P i j P_i^j P i j i i i 1 q 2 \frac{1}{q^2} q 2 1 T i T_i T i i i i
We distinguish between two cases, depending on whether the average value of T i T_i T i g ( q ) ⋅ log n g(q)\cdot\log n g ( q ) ⋅ log n g g g q q q n n n
If the average value of T i T_i T i ( 1 − 1 q ) (1-\frac{1}{q}) ( 1 − q 1 ) 1 q \frac{1}{q} q 1 1 q \frac{1}{q} q 1 1 q \frac{1}{q} q 1 T i T_i T i T i T_i T i d d d d d d
If the average value of T i T_i T i i i i W W W U U U ε n 8 \frac{\varepsilon n}{8} 8 ε n V V V V V V d d d V V V A A A
The main issue that needs to be handled in the proof is the second step of our algorithm. To use Dickson’s lemma, we used a representative supplier agent r r r U ′ U' U ′ d d d U ′ U' U ′ U U U A A A U ′ U' U ′ U ′ U' U ′ A A A U ′ U' U ′ r r r U ′ U' U ′ U ′ U' U ′ U ′ U' U ′
Related work
The subject of fair division has been extensively studied in multiple different settings. In our work, we focus specifically on indivisible goods and use MMS as our notion of fairness. For work on divisible goods, see for example [8 ] . For additional work on fair allocation of indivisible items, including envy-based notions like EF1 and EFX, see for example [4 , 1 ] . In addition, there has been a plenty of work done on fair allocations for valuations beyond additive, i.e submodular, XOS and subadditive classes from the complement free hierarchy [21 ] , which is beyond the scope of this work.
For additive valuations and every n ≥ 3 n \ge3 n ≥ 3 [20 ] . There has been a long line of work investigating the highest fraction of the MMS that can be guaranteed to each agent, and it is known to be between 7 9 \frac{7}{9} 9 7 [18 ] and 20 21 \frac{20}{21} 21 20 [13 ] . For general n n n [16 ] showed an upper bound of 1 − 1 n 4 1-\frac{1}{n^4} 1 − n 4 1 1 − Ω ( 1 log 2 n ) 1-\Omega\left(\frac{1}{\log^2 n}\right) 1 − Ω ( l o g 2 n 1 ) [13 ] .
There has also been work on MMS allocation instances where the number of items m m m n n n m ≤ n + log n m \le n+\log n m ≤ n + log n [14 ] , and may not exist when m ≤ n + c log n m \le n+c\log n m ≤ n + c log n c c c [13 ] . These works study exact MMS allocations, while our existence results consider approximate allocations in the regime m < 2 n m < 2n m < 2 n
While we consider instances with large number of agents n n n n n n n = 3 n=3 n = 3 [15 ] that 11 12 \frac{11}{12} 12 11 n = 4 n=4 n = 4 [17 ] proved an approximation ratio of 4 5 \frac{4}{5} 5 4
The concept of 1-out-of-d d d [9 ] , and later was formally introduced in [7 ] . A series of works subsequently improved the upper bound on d d d d = 4 ⌈ n / 3 ⌉ d=4\lceil n/3\rceil d = 4 ⌈ n /3 ⌉ d = n + O ( n log n ) d=n+O(\sqrt{n\log n}) d = n + O ( n log n ) [12 ] . As for the progress towards d = n + 1 d=n+1 d = n + 1 n = 4 n=4 n = 4
Regarding techniques, our algorithm utilizes the identical ordering reduction from [BL16]. When it comes to replacing MMS bundles with corresponding profiles of rounded item values, the idea is not new and has been utilized in various ways before. In [19 ] , the authors compute MMS of a single agent by representing its MMS bundles via counts of its large items in rounded value classes, and completed selected bundle types using small items. The construction somewhat resembling our head-tail profile framework was utilized in [22 ] for welfare approximations.
In the recent work [12 ] , most relevant to our topics, the authors also built their proofs using a sort of head and tail structure. However, their definition of head profiles is vastly different and is based on counts of items in certain buckets, and they control the “loads” of the buckets. Our approach instead defines head profiles based on individual value of each item, and operates with prefix loads and value breakpoints. This construction gives us much more flexibility, which is crucial to our main result.
Both of the works utilize linear programming rounding in order to obtain initial allocation for most of the agents. While the rounding of [12 ] is in some sense oblivious, our rounding algorithm is much more adaptive, which combined with our definition of head/tail profiles gives us the substantial improvement in the value of d d d d d d
In order to compensate the head profiles with tail items, [12 ] use faithful rounding method (see [5 ] ) for the common tail. Our tail completion approach is also inspired by faithful rounding, and develops its deterministic counterpart. The main distinction in our case is that tails for different agents may have different lengths, and we need to control all of them simultaneously.
It is worth mentioning that the approach of [12 ] was developed for a stronger notion of fairness called anyprice share (APS), which considers fractional partitions of items. For our results, it is crucial that we work with MMS partitions.
Technical details
We prove Theorem 1.2 first, in § 2 . The proof of Theorem 1.1 is split between Sections 3 to § 6 . In Section 3, we first consider mixed head-and-tail case when average value of tail T i T_i T i ( 1 − ε ′ ) d (1-\varepsilon')d ( 1 − ε ′ ) d § 5 , we cover the first half of the second step (adding ε ′ d \varepsilon'd ε ′ d S S S § 6 , where we describe how to use a supplier agent to add the required ε ′ d \varepsilon'd ε ′ d § 7 , we give a proof of Corollary 1.1, and mention some additional minor implications of our results. As for appendices, Appendices § 9 and § 10 cover the proofs of LP-rounding algorithms that we used in respectively § 2 and 3. Appendix § 11 gives the full proof of the supplier profile selection guarantees via Dickson’s lemma. In Appendix § 12 we give explicit constants and provide an upper bound on the claimed n ε , q n_{\varepsilon,q} n ε , q
1-out-of-d d d d = n + O ( log 2 n ) d = n + O(\log^{2} n) d = n + O ( log 2 n )
First we will prove the result on 1-out-of-d d d
Theorem 2.1 (Theorem 1.2 restated). Consider a fair allocation instance with items M M M n n n N N N { v i } i ∈ N \{v_i\}_{i\in N} { v i } i ∈ N d d d
d ≥ n + 46 ( 1 + log n ) 2 , d \ge n + 46(1+\log n)^2, d ≥ n + 46 ( 1 + log n ) 2 , In the setting of Theorem 2.1, we assume that Assumption 1 holds. In particular, we associate M M M [ m ] [m] [ m ] i ∈ N i\in N i ∈ N
v i ( 1 ) ≥ v i ( 2 ) ≥ … ≥ v i ( m ) . v_i(1) \ge v_i(2) \ge\ldots\ge v_i(m). v i ( 1 ) ≥ v i ( 2 ) ≥ … ≥ v i ( m ) . In addition, each agent i ∈ N i \in N i ∈ N d d d B i = { B i , 1 , … , B i , d } \mathcal{B}_i = \{B_{i,1},\ldots,B_{i,d}\} B i = { B i , 1 , … , B i , d } M M M v i ( B i , j ) = 1 v_i(B_{i,j}) = 1 v i ( B i , j ) = 1 j ∈ [ d ] j \in[d] j ∈ [ d ] v i ( M ) = d v_i(M) = d v i ( M ) = d i ∈ N i \in N i ∈ N
We will start with several straightforward observations.
Lemma 2.1. Let U ⊆ N U \subseteq N U ⊆ N E ⊆ M E \subseteq M E ⊆ M v u ( E ) ≥ 2 ∣ U ∣ − 1 v_u(E) \ge2|U| - 1 v u ( E ) ≥ 2∣ U ∣ − 1 u ∈ U u \in U u ∈ U E E E U U U u ∈ U u \in U u ∈ U
Proof. If ∣ U ∣ = 1 |U| = 1 ∣ U ∣ = 1 E E E U U U ∣ U ∣ > 1 |U| > 1 ∣ U ∣ > 1 B B B u ∈ U u \in U u ∈ U v u ( B ) ≥ 1 v_u(B) \ge1 v u ( B ) ≥ 1 B B B B B B u ′ ≠ u u' \ne u u ′ = u v u ′ ( B ) < 2 v_{u'}(B) < 2 v u ′ ( B ) < 2 v u ′ ( E − B ) ≥ 2 ∣ U ∣ − 3 = 2 ( ∣ U ∣ − 1 ) − 1 v_{u'}(E - B) \ge2|U| - 3 = 2(|U| - 1) - 1 v u ′ ( E − B ) ≥ 2∣ U ∣ − 3 = 2 ( ∣ U ∣ − 1 ) − 1 ∣ U ∣ |U| ∣ U ∣
If d ≥ 2 n − 1 d \ge2n - 1 d ≥ 2 n − 1 U = N U = N U = N E = M E = M E = M i ∈ N i \in N i ∈ N v i ( E ) = v i ( M ) = d ≥ 2 n − 1 = 2 ∣ U ∣ − 1 v_i(E) = v_i(M) = d \ge2n - 1 = 2|U| - 1 v i ( E ) = v i ( M ) = d ≥ 2 n − 1 = 2∣ U ∣ − 1 n + 1 ≤ d ≤ 2 n − 1 n + 1 \le d \le2n - 1 n + 1 ≤ d ≤ 2 n − 1
Lemma 2.1 suggests the following. Suppose that we allocated all but a small fraction of agents (say, n − O ( log 2 n ) n - O(\log^2 n) n − O ( log 2 n ) F ⊆ M F \subseteq M F ⊆ M F F F
Corollary 2.1. Let F ⊆ M F \subseteq M F ⊆ M v i ( F ) ≤ n + 1 v_i(F) \le n + 1 v i ( F ) ≤ n + 1 i ∈ N i \in N i ∈ N U ⊆ N U \subseteq N U ⊆ N ∣ U ∣ ≤ ( d − n ) / 2 |U| \le(d - n)/2 ∣ U ∣ ≤ ( d − n ) /2 M − F M - F M − F U U U u ∈ U u \in U u ∈ U
Proof. If U = ∅ U = \varnothing U = ∅ v u ( M ) = d v_u(M) = d v u ( M ) = d u ∈ U u \in U u ∈ U
v u ( M − F ) = v u ( M ) − v u ( F ) ≥ d − n − 1 ≥ 2 ∣ U ∣ − 1. v_u(M - F) = v_u(M) - v_u(F) \ge d - n - 1 \ge2|U| - 1. v u ( M − F ) = v u ( M ) − v u ( F ) ≥ d − n − 1 ≥ 2∣ U ∣ − 1. This is exactly the condition of Lemma 2.1 with sets E = M − F E = M - F E = M − F U U U
We will now focus on finding a set F ⊆ M F \subseteq M F ⊆ M v i ( F ) ≤ n + 1 v_i(F) \le n + 1 v i ( F ) ≤ n + 1 i ∈ N i \in N i ∈ N N ′ ⊂ N N' \subset N N ′ ⊂ N n − O ( log 2 n ) n - O(\log^2 n) n − O ( log 2 n ) N ′ N' N ′ F F F
By assumption, we associate M M M [ m ] [m] [ m ] i ∈ N i \in N i ∈ N
v i ( 1 ) ≥ v i ( 2 ) ≥ … ≥ v i ( m ) . v_i(1) \ge v_i(2) \ge\ldots\ge v_i(m). v i ( 1 ) ≥ v i ( 2 ) ≥ … ≥ v i ( m ) . We observe that, if one wants to bound the value of items F ⊆ M F \subseteq M F ⊆ M F F F prefix of M = { 1 , … , m } M = \{1,\ldots,m\} M = { 1 , … , m }
Observation 2.1. Let F ⊆ M F \subseteq M F ⊆ M 0 ≤ ℓ ≤ m 0 \le\ell\le m 0 ≤ ℓ ≤ m ∣ F ∩ [ ℓ ] ∣ ≤ η ℓ |F \cap[\ell]| \le\eta\ell ∣ F ∩ [ ℓ ] ∣ ≤ η ℓ η ∈ ( 0 , 1 ] \eta\in(0,1] η ∈ ( 0 , 1 ] i ∈ N i \in N i ∈ N v i ( F ) ≤ η d v_i(F) \le\eta d v i ( F ) ≤ η d
In particular, when η = ( n + 1 ) / d \eta= (n + 1)/d η = ( n + 1 ) / d F F F v i ( F ) ≤ n + 1 v_i(F) \le n + 1 v i ( F ) ≤ n + 1
Proof. Observe that for every 0 ≤ ℓ ≤ m 0 \le\ell\le m 0 ≤ ℓ ≤ m ℓ ∈ F \ell\in F ℓ ∈ F ∣ F ∩ [ ℓ ] ∣ − ∣ F ∩ [ ℓ − 1 ] ∣ = 1 |F \cap[\ell]| - |F \cap[\ell- 1]| = 1 ∣ F ∩ [ ℓ ] ∣ − ∣ F ∩ [ ℓ − 1 ] ∣ = 1
v i ( F ) = ∑ ℓ = 1 m v i ( ℓ ) ⋅ ( ∣ F ∩ [ ℓ ] ∣ − ∣ F ∩ [ ℓ − 1 ] ∣ ) = ∑ ℓ = 1 m ( v i ( ℓ ) − v i ( ℓ + 1 ) ) ⋅ ∣ F ∩ [ ℓ ] ∣ v_i(F) = \sum_{\ell=1}^{m} v_i(\ell) \cdot\left(|F \cap[\ell]| - |F \cap[\ell- 1]|\right) = \sum_{\ell=1}^{m} \left(v_i(\ell) - v_i(\ell+ 1)\right) \cdot|F \cap[\ell]| v i ( F ) = ℓ = 1 ∑ m v i ( ℓ ) ⋅ ( ∣ F ∩ [ ℓ ] ∣ − ∣ F ∩ [ ℓ − 1 ] ∣ ) = ℓ = 1 ∑ m ( v i ( ℓ ) − v i ( ℓ + 1 ) ) ⋅ ∣ F ∩ [ ℓ ] ∣ ≤ η ∑ ℓ = 1 m ( v i ( ℓ ) − v i ( ℓ + 1 ) ) ⋅ ℓ = η ∑ ℓ = 1 m v i ( ℓ ) = η d . \le\eta\sum_{\ell=1}^{m} \left(v_i(\ell) - v_i(\ell+ 1)\right) \cdot\ell= \eta\sum_{\ell=1}^{m} v_i(\ell) = \eta d. ≤ η ℓ = 1 ∑ m ( v i ( ℓ ) − v i ( ℓ + 1 ) ) ⋅ ℓ = η ℓ = 1 ∑ m v i ( ℓ ) = η d . We used a conventions [ 0 ] = ∅ [0] = \varnothing [ 0 ] = ∅ v i ( m + 1 ) = 0 v_i(m + 1) = 0 v i ( m + 1 ) = 0
We will refer to ∣ F ∩ [ ℓ ] ∣ |F \cap[\ell]| ∣ F ∩ [ ℓ ] ∣ ℓ \ell ℓ prefix load of the set F F F F F F M M M profiles of values of items from M M M
Demand profiles
Definition 2.1. A demand is a positive value α ∈ ( 0 , 1 ] \alpha\in(0,1] α ∈ ( 0 , 1 ] e ∈ M e \in M e ∈ M satisfies the demand α \alpha α i ∈ N i \in N i ∈ N v i ( e ) ≥ α v_i(e) \ge\alpha v i ( e ) ≥ α
Definition 2.2. A demand profile is a finite multiset π = { α 1 , … , α ∣ π ∣ } \pi= \{\alpha_1,\ldots,\alpha_{|\pi|}\} π = { α 1 , … , α ∣ π ∣ } B ⊆ M B \subseteq M B ⊆ M satisfies the demand profile π \pi π i ∈ N i \in N i ∈ N α ∈ π \alpha\in\pi α ∈ π e ∈ B e \in B e ∈ B { e 1 , … , e ∣ π ∣ } ⊆ B \{e_1,\ldots,e_{|\pi|}\} \subseteq B { e 1 , … , e ∣ π ∣ } ⊆ B
v i ( e 1 ) ≥ α 1 , … , v i ( e ∣ π ∣ ) ≥ α ∣ π ∣ . v_i(e_1) \ge\alpha_1,\ldots,v_i(e_{|\pi|}) \ge\alpha_{|\pi|}. v i ( e 1 ) ≥ α 1 , … , v i ( e ∣ π ∣ ) ≥ α ∣ π ∣ . A (guaranteed) value of a profile π \pi π
v ( π ) : = ∑ α ∈ π α . v(\pi) := \sum_{\alpha\in\pi} \alpha. v ( π ) := α ∈ π ∑ α . If agent i i i π \pi π π \pi π B ⊆ M B \subseteq M B ⊆ M v i ( B ) ≥ v ( π ) v_i(B) \ge v(\pi) v i ( B ) ≥ v ( π )
Instead of measuring a prefix load of item set F ⊆ M F \subseteq M F ⊆ M demand profile π \pi π π \pi π i i i { 1 , … , ℓ } \{1,\ldots,\ell\} { 1 , … , ℓ } i i i π \pi π breakpoints .
Definition 2.3. For i ∈ N i \in N i ∈ N α \alpha α α \alpha α breakpoint for agent i i i m i ( α ) m_i(\alpha) m i ( α ) M = [ m ] M = [m] M = [ m ] α \alpha α e ≤ m i ( α ) e \le m_i(\alpha) e ≤ m i ( α ) v i ( e ) ≥ α v_i(e) \ge\alpha v i ( e ) ≥ α
By definition, agent i i i α \alpha α { 1 , … , m i ( α ) } \{1,\ldots,m_i(\alpha)\} { 1 , … , m i ( α )} m i ( α ) ≤ ℓ m_i(\alpha) \le\ell m i ( α ) ≤ ℓ i i i { 1 , … , ℓ } \{1,\ldots,\ell\} { 1 , … , ℓ } α \alpha α ℓ \ell ℓ π \pi π π \pi π ℓ \ell ℓ
Definition 2.4. Let π = { α 1 , … , α ∣ π ∣ } \pi= \{\alpha_1,\ldots,\alpha_{|\pi|}\} π = { α 1 , … , α ∣ π ∣ } 0 ≤ ℓ ≤ m 0 \le\ell\le m 0 ≤ ℓ ≤ m i ∈ N i \in N i ∈ N P i ( π , ℓ ) \mathcal{P}_i(\pi,\ell) P i ( π , ℓ ) π \pi π ℓ \ell ℓ
P i ( π , ℓ ) = ∣ { α ∈ π : m i ( α ) ≤ ℓ } ∣ . \mathcal{P}_i(\pi,\ell) = \left|\{\alpha\in\pi: m_i(\alpha) \le\ell\}\right|. P i ( π , ℓ ) = ∣ { α ∈ π : m i ( α ) ≤ ℓ } ∣ . That is, P i ( π , ℓ ) \mathcal{P}_i(\pi,\ell) P i ( π , ℓ ) π \pi π ℓ \ell ℓ P i ( π , ℓ ) \mathcal{P}_i(\pi,\ell) P i ( π , ℓ ) ℓ \ell ℓ prefix load of π \pi π i i i
We extend the definition of prefix load to a collection of profiles.
Definition 2.5. Let I ⊆ N I \subseteq N I ⊆ N Π I : = { π i } i ∈ I \Pi_I := \{\pi_i\}_{i \in I} Π I := { π i } i ∈ I π i \pi_i π i i ∈ I i \in I i ∈ I 0 ≤ ℓ ≤ m 0 \le\ell\le m 0 ≤ ℓ ≤ m total ℓ \ell ℓ of Π I \Pi_I Π I
P ( Π I , ℓ ) : = ∑ i ∈ I P i ( π i , ℓ ) , \mathcal{P}(\Pi_I,\ell) := \sum_{i \in I} \mathcal{P}_i(\pi_i,\ell), P ( Π I , ℓ ) := i ∈ I ∑ P i ( π i , ℓ ) , the total number of demands across all agents in I I I ℓ \ell ℓ
The following lemma shows that finding an allocation of M M M I ⊆ N I \subseteq N I ⊆ N Π I \Pi_I Π I
Lemma 2.2. Let I ⊆ N I \subseteq N I ⊆ N π i = { α i , 1 , … , α i , ∣ π i ∣ } \pi_i = \{\alpha_{i,1},\ldots,\alpha_{i,|\pi_i|}\} π i = { α i , 1 , … , α i , ∣ π i ∣ } i ∈ I i \in I i ∈ I Π I : = { π i } i ∈ I \Pi_I := \{\pi_i\}_{i \in I} Π I := { π i } i ∈ I F ⊆ M F \subseteq M F ⊆ M Π I \Pi_I Π I F F F
P ( Π I , ℓ ) ≤ ∣ F ∩ [ ℓ ] ∣ , ∀ 0 ≤ ℓ ≤ m . \mathcal{P}(\Pi_I,\ell) \le|F \cap[\ell]|,\qquad\forall0 \le\ell\le m. P ( Π I , ℓ ) ≤ ∣ F ∩ [ ℓ ] ∣ , ∀0 ≤ ℓ ≤ m . Furthermore, if K : = ∑ i ∈ I ∣ π i ∣ K := \sum_{i \in I} |\pi_i| K := ∑ i ∈ I ∣ π i ∣ K K K F F F F = M F = M F = M Π I \Pi_I Π I { 1 , … , K } \{1,\ldots,K\} { 1 , … , K }
P ( Π I , ℓ ) ≤ ℓ , ∀ 0 ≤ ℓ ≤ m . \mathcal{P}(\Pi_I,\ell) \le\ell,\qquad\forall0 \le\ell\le m. P ( Π I , ℓ ) ≤ ℓ , ∀0 ≤ ℓ ≤ m . Proof. Suppose that we can satisfy all profiles of Π I \Pi_I Π I F F F
P ( Π I , ℓ ) = ∑ i ∈ I P i ( π i , ℓ ) = ∑ i ∈ I ∣ { α ∈ π i : m i ( α ) ≤ ℓ } ∣ . \mathcal{P}(\Pi_I,\ell)=\sum_{i\in I}\mathcal{P}_i(\pi_i,\ell)=\sum_{i\in I}|\{\alpha\in\pi_i:m_i(\alpha)\le\ell\}|. P ( Π I , ℓ ) = i ∈ I ∑ P i ( π i , ℓ ) = i ∈ I ∑ ∣ { α ∈ π i : m i ( α ) ≤ ℓ } ∣. Consider all α ∈ Π I \alpha\in\Pi_I α ∈ Π I ℓ \ell ℓ { 1 , … , ℓ } \{1,\ldots,\ell\} { 1 , … , ℓ } F F F ∣ F ∩ [ ℓ ] ∣ |F\cap[\ell]| ∣ F ∩ [ ℓ ] ∣ ℓ \ell ℓ P ( Π I , ℓ ) ≤ ∣ F ∩ [ ℓ ] ∣ \mathcal{P}(\Pi_I,\ell)\le|F\cap[\ell]| P ( Π I , ℓ ) ≤ ∣ F ∩ [ ℓ ] ∣
Now, suppose that P ( Π I , ℓ ) ≤ ∣ F ∩ [ ℓ ] ∣ \mathcal{P}(\Pi_I,\ell)\le|F\cap[\ell]| P ( Π I , ℓ ) ≤ ∣ F ∩ [ ℓ ] ∣ 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m ℓ = 0 \ell=0 ℓ = 0 P ( Π I , 0 ) ≤ ∣ F ∩ [ 0 ] ∣ = 0 \mathcal{P}(\Pi_I,0)\le|F\cap[0]|=0 P ( Π I , 0 ) ≤ ∣ F ∩ [ 0 ] ∣ = 0
List all individual demands α ∈ Π I \alpha\in\Pi_I α ∈ Π I ( i , α ) (i,\alpha) ( i , α ) α ∈ π i \alpha\in\pi_i α ∈ π i m i ( α ) m_i(\alpha) m i ( α ) K = ∑ i ∈ I ∣ π i ∣ = P ( Π I , m ) K=\sum_{i\in I}|\pi_i|=\mathcal{P}(\Pi_I,m) K = ∑ i ∈ I ∣ π i ∣ = P ( Π I , m )
( i 1 , α 1 ) , … , ( i K , α K ) , 1 ≤ m i 1 ( α 1 ) ≤ ⋯ ≤ m i K ( α K ) ≤ m . (i_1,\alpha_1),\ldots,(i_K,\alpha_K),\qquad1\le m_{i_1}(\alpha_1)\le\cdots\le m_{i_K}(\alpha_K)\le m. ( i 1 , α 1 ) , … , ( i K , α K ) , 1 ≤ m i 1 ( α 1 ) ≤ ⋯ ≤ m i K ( α K ) ≤ m . Note that K = P ( Π I , m ) ≤ ∣ F ∩ [ m ] ∣ K=\mathcal{P}(\Pi_I,m)\le|F\cap[m]| K = P ( Π I , m ) ≤ ∣ F ∩ [ m ] ∣ K ≤ ∣ F ∣ K\le|F| K ≤ ∣ F ∣
Denote m k : = m i k ( α k ) m_k:=m_{i_k}(\alpha_k) m k := m i k ( α k ) m 1 ≤ ⋯ ≤ m K m_1\le\cdots\le m_K m 1 ≤ ⋯ ≤ m K k k k k k k ( i , α ) (i,\alpha) ( i , α ) m i ( α ) ≤ m k m_i(\alpha)\le m_k m i ( α ) ≤ m k k k k m k m_k m k P ( Π I , m k ) ≥ k \mathcal{P}(\Pi_I,m_k)\ge k P ( Π I , m k ) ≥ k
By assumption, P ( Π I , m k ) ≤ ∣ F ∩ [ m k ] ∣ \mathcal{P}(\Pi_I,m_k)\le|F\cap[m_k]| P ( Π I , m k ) ≤ ∣ F ∩ [ m k ] ∣ F = { f 1 < ⋯ < f ∣ F ∣ } F=\{f_1<\cdots<f_{|F|}\} F = { f 1 < ⋯ < f ∣ F ∣ } f k ≤ m k f_k\le m_k f k ≤ m k m k = m i k ( α k ) m_k=m_{i_k}(\alpha_k) m k = m i k ( α k ) f k f_k f k i k i_k i k α k \alpha_k α k m k ≥ f k m_k\ge f_k m k ≥ f k v i k ( f k ) ≥ α k v_{i_k}(f_k)\ge\alpha_k v i k ( f k ) ≥ α k 1 ≤ k ≤ K 1\le k\le K 1 ≤ k ≤ K K ≤ ∣ F ∣ K\le|F| K ≤ ∣ F ∣ K K K F F F □ \square □
Recall that in Observation 2.1, however, we seek for a set F ⊆ M F\subseteq M F ⊆ M F F F ∣ F ∩ [ ℓ ] ∣ ≤ η ℓ |F\cap[\ell]|\le\eta\ell ∣ F ∩ [ ℓ ] ∣ ≤ η ℓ ℓ \ell ℓ F ⊆ M F\subseteq M F ⊆ M Π I \Pi_I Π I
Corollary 2.2 (Refinement of Lemma 2.2). Let I ⊆ N I\subseteq\mathbb{N} I ⊆ N π i \pi_i π i i ∈ I i\in I i ∈ I Π I : = { π i } i ∈ I \Pi_I:=\{\pi_i\}_{i\in I} Π I := { π i } i ∈ I η ∈ ( 0 , 1 ] \eta\in(0,1] η ∈ ( 0 , 1 ] 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m
P ( Π I , ℓ ) ≤ η ℓ . \mathcal{P}(\Pi_I,\ell)\le\eta\ell. P ( Π I , ℓ ) ≤ η ℓ . Then, all profiles of Π I \Pi_I Π I F ⊆ M F\subseteq M F ⊆ M
∣ F ∩ [ ℓ ] ∣ ≤ η ℓ , ∀ 0 ≤ ℓ ≤ m . |F\cap[\ell]|\le\eta\ell,\qquad\forall0\le\ell\le m. ∣ F ∩ [ ℓ ] ∣ ≤ η ℓ , ∀0 ≤ ℓ ≤ m . Proof. List all demands α ∈ Π I \alpha\in\Pi_I α ∈ Π I ( i , α ) (i,\alpha) ( i , α ) α ∈ π i \alpha\in\pi_i α ∈ π i m i ( α ) m_i(\alpha) m i ( α ) K = ∑ i ∈ I ∣ π i ∣ = P ( Π I , m ) K=\sum_{i\in I}|\pi_i|=\mathcal{P}(\Pi_I,m) K = ∑ i ∈ I ∣ π i ∣ = P ( Π I , m )
( i 1 , α 1 ) , … , ( i K , α K ) , 1 ≤ m i 1 ( α 1 ) ≤ ⋯ ≤ m i K ( α K ) ≤ m . (i_1,\alpha_1),\ldots,(i_K,\alpha_K),\qquad1\le m_{i_1}(\alpha_1)\le\cdots\le m_{i_K}(\alpha_K)\le m. ( i 1 , α 1 ) , … , ( i K , α K ) , 1 ≤ m i 1 ( α 1 ) ≤ ⋯ ≤ m i K ( α K ) ≤ m . Denote m k : = m i k ( α k ) m_k:=m_{i_k}(\alpha_k) m k := m i k ( α k ) m 1 ≤ ⋯ ≤ m K m_1\le\cdots\le m_K m 1 ≤ ⋯ ≤ m K k ∈ [ K ] k\in[K] k ∈ [ K ] k ≤ P ( Π I , m k ) ≤ η m k k\le\mathcal{P}(\Pi_I,m_k)\le\eta m_k k ≤ P ( Π I , m k ) ≤ η m k m k = m i k ( α k ) m_k=m_{i_k}(\alpha_k) m k = m i k ( α k ) f k : = ⌈ k / η ⌉ f_k:=\lceil k/\eta\rceil f k := ⌈ k / η ⌉ i k i_k i k 1 ≤ f k ≤ m k ≤ m 1\le f_k\le m_k\le m 1 ≤ f k ≤ m k ≤ m f k ∈ M f_k\in M f k ∈ M α k \alpha_k α k η ≤ 1 \eta\le1 η ≤ 1 f k + 1 ≥ f k + 1 f_{k+1}\ge f_k+1 f k + 1 ≥ f k + 1
Let F = { f k } k ∈ [ K ] F=\{f_k\}_{k\in[K]} F = { f k } k ∈ [ K ] 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m
∣ F ∩ [ ℓ ] ∣ = ∣ { k ∈ [ K ] : ⌈ k / η ⌉ ≤ ℓ } ∣ = ∣ { k ∈ [ K ] : k ≤ η ℓ } ∣ = min { K , ⌊ η ℓ ⌋ } ≤ η ℓ . |F\cap[\ell]|=|\{k\in[K]:\lceil k/\eta\rceil\le\ell\}|=|\{k\in[K]:k\le\eta\ell\}|=\min\{K,\lfloor\eta\ell\rfloor\}\le\eta\ell. ∣ F ∩ [ ℓ ] ∣ = ∣ { k ∈ [ K ] : ⌈ k / η ⌉ ≤ ℓ } ∣ = ∣ { k ∈ [ K ] : k ≤ η ℓ } ∣ = min { K , ⌊ η ℓ ⌋} ≤ η ℓ . □ \square □
Head and tail profiles So, to find a set F ⊆ M F\subseteq M F ⊆ M Π I \Pi_I Π I I ⊆ N I\subseteq\mathbb{N} I ⊆ N i ∈ N i\in\mathbb{N} i ∈ N B i = { B i , 1 , … , B i , d } \mathcal{B}_i=\{B_{i,1},\ldots,B_{i,d}\} B i = { B i , 1 , … , B i , d } π i , j = { v i ( e ) : e ∈ B i , j and v i ( e ) > 0 } \pi_{i,j}=\{v_i(e):e\in B_{i,j}\text{ and }v_i(e)>0\} π i , j = { v i ( e ) : e ∈ B i , j and v i ( e ) > 0 } π i , j \pi_{i,j} π i , j m m m π \pi π head profile with bounded number of demands, and a tail profile .
Head and tail profiles
As per Corollary 2.2, we would like to find a set I ⊆ N I \subseteq N I ⊆ N ∣ I ∣ |I| ∣ I ∣
Instead of finding a collection of full profiles of value 1 for some subset I I I Π I \Pi_I Π I head profiles π i ∈ Π I \pi_i \in\Pi_I π i ∈ Π I Ψ I \Psi_I Ψ I tail profiles ψ i ∈ Ψ I \psi_i \in\Psi_I ψ i ∈ Ψ I i ∈ I i \in I i ∈ I { π i , ψ i } \{\pi_i,\psi_i\} { π i , ψ i }
Definition 2.6. For h ∈ [ m ] h \in[m] h ∈ [ m ] { 1 , … , h } \{1,\ldots,h\} { 1 , … , h } head items , and { h + 1 , … , m } \{h+1,\ldots,m\} { h + 1 , … , m } tail items . For agent i ∈ N i \in N i ∈ N tail value of i i i
T i = ∑ e > h v i ( e ) . T_i = \sum_{e>h} v_i(e). T i = e > h ∑ v i ( e ) . Definition 2.7. For an agent i ∈ N i \in N i ∈ N π \pi π head profile for i i i α ∈ π \alpha\in\pi α ∈ π α = v i ( e ) \alpha= v_i(e) α = v i ( e ) head item e ≤ h e \le h e ≤ h ψ \psi ψ tail profile for i i i β ∈ ψ \beta\in\psi β ∈ ψ β = v i ( e ) \beta= v_i(e) β = v i ( e ) tail item e > h e > h e > h
We will also need two additional conventions.
Definition 2.8. An empty profile is ∅ \varnothing ∅ i ∈ N i \in N i ∈ N π = ∅ \pi= \varnothing π = ∅ ψ = ∅ \psi= \varnothing ψ = ∅ P i ( ∅ , ℓ ) = 0 \mathcal{P}_i(\varnothing,\ell) = 0 P i ( ∅ , ℓ ) = 0 0 ≤ ℓ ≤ m 0 \le\ell\le m 0 ≤ ℓ ≤ m
We allow an agent i ∈ N i \in N i ∈ N I I I π i = ∅ \pi_i = \varnothing π i = ∅ ψ i = ∅ \psi_i = \varnothing ψ i = ∅
We will also define a “no-profile” ⊥ \bot ⊥ i ∈ N i \in N i ∈ N I I I
Definition 2.9. A no-profile is ⊥ \bot ⊥ i ∈ N i \in N i ∈ N ⊥ \bot ⊥ P i ( ⊥ , ℓ ) = 0 \mathcal{P}_i(\bot,\ell) = 0 P i ( ⊥ , ℓ ) = 0 0 ≤ ℓ ≤ m 0 \le\ell\le m 0 ≤ ℓ ≤ m
Our approach is to first find a sufficiently large set I ⊆ N I \subseteq N I ⊆ N head profiles Π I = { π i } i ∈ I \Pi_I = \{\pi_i\}_{i\in I} Π I = { π i } i ∈ I [ h ] [h] [ h ] N − I N-I N − I ⊥ \bot ⊥ i ∈ I i \in I i ∈ I π i \pi_i π i ψ i \psi_i ψ i v ( π i ) + v ( ψ i ) ≥ 1 v(\pi_i) + v(\psi_i) \ge1 v ( π i ) + v ( ψ i ) ≥ 1 i i i
Since profiles Π I \Pi_I Π I M M M π i ∈ Π I \pi_i \in\Pi_I π i ∈ Π I v ( π i ) < 1 v(\pi_i) < 1 v ( π i ) < 1 i ∈ I i \in I i ∈ I π i \pi_i π i
Definition 2.10. For a head profile π ≠ ⊥ \pi\ne\bot π = ⊥ deficit is
Δ ( π ) : = ( 1 − v ( π ) ) + . \Delta(\pi) := (1-v(\pi))_+. Δ ( π ) := ( 1 − v ( π ) ) + . Here ( ⋅ ) + : = max { 0 , ⋅ } (\cdot)_+ := \max\{0,\cdot\} ( ⋅ ) + := max { 0 , ⋅ }
Suppose that agent i ∈ N i \in N i ∈ N π i ≠ ⊥ \pi_i \ne\bot π i = ⊥ Δ ( π i ) > 0 \Delta(\pi_i) > 0 Δ ( π i ) > 0 π i \pi_i π i [ h ] [h] [ h ] i i i { h + 1 , … , m } \{h+1,\ldots,m\} { h + 1 , … , m }
We do so using an approach inspired by faithful rounding method, which also appeared in [5 , 12 ] . Suppose that, in addition to head items satisfying π i \pi_i π i i i i Δ ( π i ) T i \frac{\Delta(\pi_i)}{T_i} T i Δ ( π i ) e > h e > h e > h i i i
v ( π i ) + Δ ( π i ) T i ∑ e > h v i ( e ) = v ( π i ) + Δ ( π i ) ≥ 1. v(\pi_i) + \frac{\Delta(\pi_i)}{T_i}\sum_{e>h}v_i(e) = v(\pi_i) + \Delta(\pi_i) \ge1. v ( π i ) + T i Δ ( π i ) e > h ∑ v i ( e ) = v ( π i ) + Δ ( π i ) ≥ 1. In essence, value Δ ( π i ) T i \frac{\Delta(\pi_i)}{T_i} T i Δ ( π i ) i i i π i \pi_i π i
Definition 2.11. For agent i ∈ N i \in N i ∈ N π \pi π
τ i ( π ) : = { Δ ( π ) T i , π ≠ ⊥ and T i > 0 ; 0 , otherwise. \tau_i(\pi) :=
\begin{cases}
\frac{\Delta(\pi)}{T_i}, & \pi\ne\perp\text{ and } T_i > 0;\\
0, & \text{otherwise.}
\end{cases} τ i ( π ) := { T i Δ ( π ) , 0 , π = ⊥ and T i > 0 ; otherwise. Consider some prefix ℓ > h \ell> h ℓ > h τ i ( π i ) \tau_i(\pi_i) τ i ( π i ) e > h e > h e > h τ i ( π i ) \tau_i(\pi_i) τ i ( π i ) ℓ > h \ell> h ℓ > h i i i τ i ( π i ) ( ℓ − h ) \tau_i(\pi_i)(\ell-h) τ i ( π i ) ( ℓ − h )
Lemma 2.3 (Tail completion). Let I ⊆ N I \subseteq N I ⊆ N π i \pi_i π i i ∈ I i \in I i ∈ I Π I = { π i } i ∈ I \Pi_I = \{\pi_i\}_{i\in I} Π I = { π i } i ∈ I T i = 0 T_i = 0 T i = 0 Δ ( π i ) = 0 \Delta(\pi_i) = 0 Δ ( π i ) = 0 i ∈ I i \in I i ∈ I η ∈ ( 0 , 1 ] \eta\in(0,1] η ∈ ( 0 , 1 ] 0 ≤ ℓ ≤ h 0 \leq\ell\leq h 0 ≤ ℓ ≤ h P ( Π I , ℓ ) ≤ η ℓ \mathcal{P}(\Pi_I,\ell) \leq\eta\ell P ( Π I , ℓ ) ≤ η ℓ h + 1 ≤ ℓ ≤ m h+1 \leq\ell\leq m h + 1 ≤ ℓ ≤ m
P ( Π I , ℓ ) ≤ η ℓ − ( ℓ − h ) ∑ i ∈ I τ i ( π i ) − ∣ I ∣ . \mathcal{P}(\Pi_I,\ell) \leq\eta\ell- (\ell-h)\sum_{i\in I}\tau_i(\pi_i) - |I|. P ( Π I , ℓ ) ≤ η ℓ − ( ℓ − h ) i ∈ I ∑ τ i ( π i ) − ∣ I ∣. There exists a collection Ψ I : = { ψ i } i ∈ I \Psi_I := \{\psi_i\}_{i\in I} Ψ I := { ψ i } i ∈ I i i i ψ i \psi_i ψ i
∀ 0 ≤ ℓ ≤ m : P ( Π I , ℓ ) + P ( Ψ I , ℓ ) ≤ η ℓ , \forall0 \leq\ell\leq m:\quad\mathcal{P}(\Pi_I,\ell) + \mathcal{P}(\Psi_I,\ell) \leq\eta\ell, ∀0 ≤ ℓ ≤ m : P ( Π I , ℓ ) + P ( Ψ I , ℓ ) ≤ η ℓ , and for every i ∈ I i \in I i ∈ I
Proof. Consider i ∈ I i \in I i ∈ I τ i : = τ i ( π i ) \tau_i := \tau_i(\pi_i) τ i := τ i ( π i ) ψ i \psi_i ψ i { h + 1 , … , m } \{h+1,\ldots,m\} { h + 1 , … , m } 1 / τ i 1/\tau_i 1/ τ i ψ i \psi_i ψ i t = 1 , … , ⌈ τ i ⋅ ( m − h ) ⌉ t = 1,\ldots,\lceil\tau_i\cdot(m-h)\rceil t = 1 , … , ⌈ τ i ⋅ ( m − h )⌉ v i ( e i , t ) v_i(e_{i,t}) v i ( e i , t ) ψ i \psi_i ψ i
e i , t = h + 1 + ⌊ t − 1 τ i ⌋ . e_{i,t}=h+1+\left\lfloor\frac{t-1}{\tau_i}\right\rfloor. e i , t = h + 1 + ⌊ τ i t − 1 ⌋ . The intuition behind this construction is straightforward: after taking the first item of every segment of length 1 / τ i 1/\tau_i 1/ τ i τ i \tau_i τ i τ i = 0 \tau_i = 0 τ i = 0 ψ i = ∅ \psi_i = \varnothing ψ i = ∅
We can describe this construction equivalently as follows. For 1 ≤ k ≤ m − h 1 \leq k \leq m-h 1 ≤ k ≤ m − h β i , k : = v i ( h + k ) \beta_{i,k}:=v_i(h+k) β i , k := v i ( h + k ) k k k i i i k ∈ [ m − h ] k \in[m-h] k ∈ [ m − h ]
⌈ τ i ⋅ k ⌉ − ⌈ τ i ⋅ ( k − 1 ) ⌉ \lceil\tau_i\cdot k\rceil-\lceil\tau_i\cdot(k-1)\rceil ⌈ τ i ⋅ k ⌉ − ⌈ τ i ⋅ ( k − 1 )⌉ copies of β i , k \beta_{i,k} β i , k ψ i \psi_i ψ i
e i , t ≤ h + k ⟺ ⌊ t − 1 τ i ⌋ ≤ k − 1 ⟺ t − 1 < τ i ⋅ k ⟺ t ≤ ⌈ τ i ⋅ k ⌉ . e_{i,t}\leq h+k \Longleftrightarrow\left\lfloor\frac{t-1}{\tau_i}\right\rfloor\leq k-1 \Longleftrightarrow t-1<\tau_i\cdot k \Longleftrightarrow t\leq\lceil\tau_i\cdot k\rceil. e i , t ≤ h + k ⟺ ⌊ τ i t − 1 ⌋ ≤ k − 1 ⟺ t − 1 < τ i ⋅ k ⟺ t ≤ ⌈ τ i ⋅ k ⌉ . So, the first k k k ⌈ τ i ⋅ k ⌉ \lceil\tau_i\cdot k\rceil ⌈ τ i ⋅ k ⌉ ψ i \psi_i ψ i k k k ⌈ τ i ⋅ k ⌉ − ⌈ τ i ⋅ ( k − 1 ) ⌉ \lceil\tau_i\cdot k\rceil-\lceil\tau_i\cdot(k-1)\rceil ⌈ τ i ⋅ k ⌉ − ⌈ τ i ⋅ ( k − 1 )⌉ τ i > 1 \tau_i>1 τ i > 1 i i i ψ i \psi_i ψ i τ i ≤ 1 \tau_i\leq1 τ i ≤ 1 ψ i \psi_i ψ i
It is easy to see that such choice of ψ i \psi_i ψ i i i i
v ( ψ i ) = ∑ k = 1 m − h β i , k ⋅ ( ⌈ τ i ⋅ k ⌉ − ⌈ τ i ⋅ ( k − 1 ) ⌉ ) = ∑ k = 1 m − h ⌈ τ i ⋅ k ⌉ ⋅ ( β i , k − β i , k + 1 ) ≥ τ i ∑ k = 1 m − h k ⋅ ( β i , k − β i , k + 1 ) = τ i ∑ k = 1 m − h β i , k = τ i T i = Δ ( π i ) . \begin{aligned}
v(\psi_i) = \sum_{k=1}^{m-h} \beta_{i,k} \cdot\left(\lceil\tau_i \cdot k \rceil- \lceil\tau_i \cdot(k-1) \rceil\right) = \sum_{k=1}^{m-h} \lceil\tau_i \cdot k \rceil\cdot\left(\beta_{i,k} - \beta_{i,k+1}\right) \\
&\ge\tau_i \sum_{k=1}^{m-h} k \cdot\left(\beta_{i,k} - \beta_{i,k+1}\right) = \tau_i \sum_{k=1}^{m-h} \beta_{i,k} = \tau_i T_i = \Delta(\pi_i).
\end{aligned} v ( ψ i ) = k = 1 ∑ m − h β i , k ⋅ ( ⌈ τ i ⋅ k ⌉ − ⌈ τ i ⋅ ( k − 1 )⌉ ) = k = 1 ∑ m − h ⌈ τ i ⋅ k ⌉ ⋅ ( β i , k − β i , k + 1 ) ≥ τ i k = 1 ∑ m − h k ⋅ ( β i , k − β i , k + 1 ) = τ i k = 1 ∑ m − h β i , k = τ i T i = Δ ( π i ) . Since Δ ( π i ) = ( 1 − v ( π i ) ) + \Delta(\pi_i) = (1-v(\pi_i))_+ Δ ( π i ) = ( 1 − v ( π i ) ) + v ( π i ) + ψ ( π i ) ≥ 1 v(\pi_i) + \psi(\pi_i) \ge1 v ( π i ) + ψ ( π i ) ≥ 1
It remains to bound the additional prefix load generated by Ψ I = { ψ i } i ∈ I \Psi_I = \{\psi_i\}_{i\in I} Ψ I = { ψ i } i ∈ I i ∈ I i \in I i ∈ I β i , k ∈ ψ i \beta_{i,k} \in\psi_i β i , k ∈ ψ i h + k h+k h + k 0 ≤ ℓ ≤ h 0 \le\ell\le h 0 ≤ ℓ ≤ h
P i ( ψ i , ℓ ) = ∑ β ∈ ψ i 1 { m i ( β ) ≤ ℓ } = 0. \mathcal{P}_i(\psi_i,\ell) = \sum_{\beta\in\psi_i} \mathbf{1}\{m_i(\beta) \le\ell\} = 0. P i ( ψ i , ℓ ) = β ∈ ψ i ∑ 1 { m i ( β ) ≤ ℓ } = 0. for every i ∈ I i \in I i ∈ I P ( Ψ I , ℓ ) = 0 \mathcal{P}(\Psi_I,\ell) = 0 P ( Ψ I , ℓ ) = 0 ℓ > h \ell> h ℓ > h ψ i \psi_i ψ i k = ℓ − h k=\ell-h k = ℓ − h ⌈ τ i ⋅ ( ℓ − h ) ⌉ ≤ τ i ⋅ ( ℓ − h ) + 1 \lceil\tau_i\cdot(\ell-h)\rceil\le\tau_i\cdot(\ell-h)+1 ⌈ τ i ⋅ ( ℓ − h )⌉ ≤ τ i ⋅ ( ℓ − h ) + 1 Ψ I \Psi_I Ψ I ℓ > h \ell> h ℓ > h
P ( Ψ I , ℓ ) ≤ ∑ i ∈ I ( τ i ⋅ ( ℓ − h ) + 1 ) = ( ℓ − h ) ∑ i ∈ I τ i + ∣ I ∣ . \mathcal{P}(\Psi_I,\ell) \le\sum_{i\in I}(\tau_i\cdot(\ell-h)+1) = (\ell-h)\sum_{i\in I}\tau_i + |I|. P ( Ψ I , ℓ ) ≤ i ∈ I ∑ ( τ i ⋅ ( ℓ − h ) + 1 ) = ( ℓ − h ) i ∈ I ∑ τ i + ∣ I ∣. The lemma follows.
Thus, in order to get a set I ⊆ N I \subseteq N I ⊆ N full profiles satisfying Corollary 2.2, it suffices to find I ⊆ N I \subseteq N I ⊆ N only head profiles Π I = { π i } i ∈ I \Pi_I=\{\pi_i\}_{i\in I} Π I = { π i } i ∈ I I ⊆ N I \subseteq N I ⊆ N n − O ( log 2 n ) n-O(\log^2 n) n − O ( log 2 n ) Π I \Pi_I Π I
Fractional head profiles
We will first define a collection of head profiles that agents i ∈ N i \in N i ∈ N
Definition 2.12. For every agent i ∈ N i \in N i ∈ N B i = { B i , 1 , … , B i , d } \mathcal{B}_i=\{B_{i,1},\ldots,B_{i,d}\} B i = { B i , 1 , … , B i , d } d d d M M M MMS ( M , v i , d ) \operatorname{MMS}(M,v_i,d) MMS ( M , v i , d ) B i , j ∈ B i B_{i,j}\in\mathcal{B}_i B i , j ∈ B i v i ( B i , j ) = 1 v_i(B_{i,j})=1 v i ( B i , j ) = 1 j ∈ [ d ] j\in[d] j ∈ [ d ]
π i , j : = { v i ( e ) : e ∈ B i , j ∩ [ h ] , v i ( e ) > 0 } . \pi_{i,j}:=\{v_i(e):e\in B_{i,j}\cap[h],v_i(e)>0\}. π i , j := { v i ( e ) : e ∈ B i , j ∩ [ h ] , v i ( e ) > 0 } . Let Π : = { ∅ , ⊥ } + ⋃ i ∈ N { π i , j : j ∈ [ d ] } \Pi:=\{\varnothing,\bot\}+\bigcup_{i\in N}\{\pi_{i,j}:j\in[d]\} Π := { ∅ , ⊥ } + ⋃ i ∈ N { π i , j : j ∈ [ d ]}
Instead of considering these profiles directly, for every agent i ∈ N i \in N i ∈ N distribution over profiles Π \Pi Π fractional profiles . Our goal then will be first to define a collection of fractional profiles that satisfy the prefix load condition of Lemma 2.3, and then round it to an integral collection while preserving every prefix constraint.
Definition 2.13. Let Π \Pi Π collection of head profiles , containing ∅ \varnothing ∅ ⊥ \bot ⊥ fractional head profile of agent i ∈ N i \in N i ∈ N ( x i , Π ) (x_i,\Pi) ( x i , Π ) x i ( π ) ≥ 0 x_i(\pi)\ge0 x i ( π ) ≥ 0 π ∈ Π \pi\in\Pi π ∈ Π ∑ π ∈ Π x i ( π ) = 1 \sum_{\pi\in\Pi}x_i(\pi)=1 ∑ π ∈ Π x i ( π ) = 1
We naturally extend the definition of (head) prefix load to fractional profiles.
Definition 2.14. For agent i ∈ N i \in N i ∈ N fractional head profile ( x i , Π ) (x_i,\Pi) ( x i , Π ) fractional ℓ \ell ℓ of Π \Pi Π 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m
H i ( x i , Π , ℓ ) : = ∑ π ∈ Π x i ( π ) ⋅ P i ( π , ℓ ) . \mathcal{H}_i(x_i,\Pi,\ell):=\sum_{\pi\in\Pi}x_i(\pi)\cdot\mathcal{P}_i(\pi,\ell). H i ( x i , Π , ℓ ) := π ∈ Π ∑ x i ( π ) ⋅ P i ( π , ℓ ) . Assume that every agent i ∈ N i \in N i ∈ N ( x i , Π ) (x_i,\Pi) ( x i , Π ) Π \Pi Π x = { x i } i ∈ N x=\{x_i\}_{i\in N} x = { x i } i ∈ N 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m total fractional ℓ \ell ℓ of Π \Pi Π x x x
H ( x , Π , ℓ ) : = ∑ i ∈ N H i ( x i , Π , ℓ ) = ∑ i ∈ N ∑ π ∈ Π x i ( π ) ⋅ P i ( π , ℓ ) . \mathcal{H}(x,\Pi,\ell):=\sum_{i\in N}\mathcal{H}_i(x_i,\Pi,\ell)=\sum_{i\in N}\sum_{\pi\in\Pi}x_i(\pi)\cdot\mathcal{P}_i(\pi,\ell). H ( x , Π , ℓ ) := i ∈ N ∑ H i ( x i , Π , ℓ ) = i ∈ N ∑ π ∈ Π ∑ x i ( π ) ⋅ P i ( π , ℓ ) . In addition to the fractional head load, we will also consider the fractional tail load . As all agents share a common tail { h + 1 , … , m } \{h+1,\ldots,m\} { h + 1 , … , m } ℓ \ell ℓ ( ℓ − h ) (\ell-h) ( ℓ − h )
Definition 2.15. For agent i ∈ N i \in N i ∈ N fractional head profile ( x i , Π ) (x_i,\Pi) ( x i , Π ) fractional tail load of Π \Pi Π
τ i ( x i , Π ) : = ∑ π ∈ Π x i ( π ) ⋅ τ i ( π ) . \tau_i(x_i,\Pi) := \sum_{\pi\in\Pi} x_i(\pi)\cdot\tau_i(\pi). τ i ( x i , Π ) := π ∈ Π ∑ x i ( π ) ⋅ τ i ( π ) . Assume that every agent i ∈ N i \in N i ∈ N ( x i , Π ) (x_i,\Pi) ( x i , Π ) Π \Pi Π x = { x i } i ∈ N x=\{x_i\}_{i\in N} x = { x i } i ∈ N total fractional tail load of Π \Pi Π x x x
τ ( x , Π ) : = ∑ i ∈ N τ i ( x i , Π ) = ∑ i ∈ N ∑ π ∈ Π x i ( π ) ⋅ τ i ( π ) . \tau(x,\Pi):=\sum_{i\in N}\tau_i(x_i,\Pi)=\sum_{i\in N}\sum_{\pi\in\Pi}x_i(\pi)\cdot\tau_i(\pi). τ ( x , Π ) := i ∈ N ∑ τ i ( x i , Π ) = i ∈ N ∑ π ∈ Π ∑ x i ( π ) ⋅ τ i ( π ) . Finally, we will be interested in the expected number of agents i ∈ N i\in N i ∈ N ⊥ \perp ⊥ ( x i , Π ) (x_i,\Pi) ( x i , Π )
Definition 2.16. Assume that every agent i ∈ N i\in N i ∈ N ( x i , Π ) (x_i,\Pi) ( x i , Π ) Π \Pi Π x = { x i } i ∈ N x=\{x_i\}_{i\in N} x = { x i } i ∈ N fractional number of assigned profiles with respect to ( x , Π ) (x,\Pi) ( x , Π )
I ( x , Π ) : = ∑ i ∈ N ∑ π ≠ ⊥ x i ( π ) . I(x,\Pi):=\sum_{i\in N}\sum_{\pi\ne\perp}x_i(\pi). I ( x , Π ) := i ∈ N ∑ π = ⊥ ∑ x i ( π ) . For every agent i ∈ N i\in N i ∈ N ( x i , Π ) (x_i,\Pi) ( x i , Π ) i i i
Definition 2.17. Let Π : = { ∅ , ⊥ } + ⋃ i ∈ N { π i , j : j ∈ [ d ] } \Pi:=\{\varnothing,\perp\}+\bigcup_{i\in N}\{\pi_{i,j}:j\in[d]\} Π := { ∅ , ⊥ } + ⋃ i ∈ N { π i , j : j ∈ [ d ]} i ∈ N i\in N i ∈ N ( x i ∗ , Π ) (x_i^*,\Pi) ( x i ∗ , Π ) π ∈ Π \pi\in\Pi π ∈ Π
x i ∗ ( π ) = ∣ { j ∈ [ d ] : π i , j = π } ∣ d . x_i^*(\pi)=\frac{|\{j\in[d]:\pi_{i,j}=\pi\}|}{d}. x i ∗ ( π ) = d ∣ { j ∈ [ d ] : π i , j = π } ∣ . We will refer to ( x i ∗ , Π ) (x_i^*,\Pi) ( x i ∗ , Π ) fractional MMS head profile of agent i i i
We will now show that the MMS profiles π i , j \pi_{i,j} π i , j i ∈ N i\in N i ∈ N ( x i ∗ , Π ) (x_i^*,\Pi) ( x i ∗ , Π )
Lemma 2.4. Consider the fractional MMS head profiles ( x ∗ , Π ) (x^*,\Pi) ( x ∗ , Π ) i ∈ N i\in N i ∈ N
∀ 0 ≤ ℓ ≤ m : H ( x ∗ , Π , ℓ ) ≤ n d min { ℓ , h } ; τ ( x ∗ , Π ) ≤ n d , I ( x ∗ , Π ) = n . \forall0\le\ell\le m:\ \mathcal{H}(x^*,\Pi,\ell)\le\frac{n}{d}\min\{\ell,h\};\qquad\tau(x^*,\Pi)\le\frac{n}{d},\qquad I(x^*,\Pi)=n. ∀0 ≤ ℓ ≤ m : H ( x ∗ , Π , ℓ ) ≤ d n min { ℓ , h } ; τ ( x ∗ , Π ) ≤ d n , I ( x ∗ , Π ) = n . As a corollary, for η = ( n + 1 ) / d \eta=(n+1)/d η = ( n + 1 ) / d h = min { n d , m } h=\min\{nd,m\} h = min { n d , m } 0 ≤ ℓ ≤ h 0\le\ell\le h 0 ≤ ℓ ≤ h H ( x ∗ , Π , ℓ ) ≤ η ℓ \mathcal{H}(x^*,\Pi,\ell)\le\eta\ell H ( x ∗ , Π , ℓ ) ≤ η ℓ h + 1 ≤ ℓ ≤ m h+1\le\ell\le m h + 1 ≤ ℓ ≤ m
H ( x ∗ , Π , ℓ ) ≤ η ℓ − ( ℓ − h ) ⋅ τ ( x ∗ , Π ) − I ( x ∗ , Π ) . \mathcal{H}(x^*,\Pi,\ell)\le\eta\ell-(\ell-h)\cdot\tau(x^*,\Pi)-I(x^*,\Pi). H ( x ∗ , Π , ℓ ) ≤ η ℓ − ( ℓ − h ) ⋅ τ ( x ∗ , Π ) − I ( x ∗ , Π ) . Proof. We first observe that for every agent i ∈ N i\in N i ∈ N 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m
∑ j = 1 d P i ( π i , j , ℓ ) ≤ min { ℓ , h } . \sum_{j=1}^{d}\mathcal{P}_i(\pi_{i,j},\ell)\le\min\{\ell,h\}. j = 1 ∑ d P i ( π i , j , ℓ ) ≤ min { ℓ , h } . By definition P i ( π i , j , ℓ ) \mathcal{P}_i(\pi_{i,j},\ell) P i ( π i , j , ℓ ) π i , j \pi_{i,j} π i , j ℓ \ell ℓ i i i π i , j \pi_{i,j} π i , j B i , j B_{i,j} B i , j B i = { B i , 1 , … , B i , d } B_i=\{B_{i,1},\ldots,B_{i,d}\} B i = { B i , 1 , … , B i , d } ℓ \ell ℓ ℓ \ell ℓ ℓ \ell ℓ
{ π i , 1 , … , π i , d } \{\pi_{i,1},\ldots,\pi_{i,d}\} { π i , 1 , … , π i , d } π i , j \pi_{i,j} π i , j [ h ] [h] [ h ] h h h i i i ∑ j = 1 d P i ( π i , j , ℓ ) ≤ min ( ℓ , h ) \sum_{j=1}^{d}\mathcal{P}_i(\pi_{i,j},\ell)\leq\min(\ell,h) ∑ j = 1 d P i ( π i , j , ℓ ) ≤ min ( ℓ , h )
It immediately follows that
H ( x ∗ , Π , ℓ ) = 1 d ∑ i ∈ N ∑ j = 1 d P i ( π i , j , ℓ ) ≤ n d min { ℓ , h } . \mathcal{H}(x^*,\Pi,\ell)=\frac{1}{d}\sum_{i\in N}\sum_{j=1}^{d}\mathcal{P}_i(\pi_{i,j},\ell)\leq\frac{n}{d}\min\{\ell,h\}. H ( x ∗ , Π , ℓ ) = d 1 i ∈ N ∑ j = 1 ∑ d P i ( π i , j , ℓ ) ≤ d n min { ℓ , h } . Next, for every i ∈ N i\in N i ∈ N j ∈ [ d ] j\in[d] j ∈ [ d ]
Δ ( π i , j ) = 1 − v ( π i , j ) = v ( B i , j ∩ { h + 1 , … , m } ) , \Delta(\pi_{i,j})=1-v(\pi_{i,j})=v(B_{i,j}\cap\{h+1,\ldots,m\}), Δ ( π i , j ) = 1 − v ( π i , j ) = v ( B i , j ∩ { h + 1 , … , m }) , so by additivity it holds ∑ j = 1 d Δ ( π i , j ) = T i \sum_{j=1}^{d}\Delta(\pi_{i,j})=T_i ∑ j = 1 d Δ ( π i , j ) = T i
τ ( x ∗ , Π ) = 1 d ∑ i ∈ N T i > 0 ∑ j = 1 d Δ ( π i , j ) T i ≤ n d . \tau(x^*,\Pi)=\frac{1}{d}\sum_{\substack{i\in N\\T_i>0}}\sum_{j=1}^{d}\frac{\Delta(\pi_{i,j})}{T_i}\leq\frac{n}{d}. τ ( x ∗ , Π ) = d 1 i ∈ N T i > 0 ∑ j = 1 ∑ d T i Δ ( π i , j ) ≤ d n . The equality I ( x ∗ , Π ) = n \mathcal{I}(x^*,\Pi)=n I ( x ∗ , Π ) = n ∑ π ≠ ⊥ x i ∗ ( π i , j ) = 1 \sum_{\pi\ne\perp}x_i^*(\pi_{i,j})=1 ∑ π = ⊥ x i ∗ ( π i , j ) = 1 i ∈ N i\in N i ∈ N η = ( n + 1 ) / d \eta=(n+1)/d η = ( n + 1 ) / d h = min { n d , m } h=\min\{nd,m\} h = min { n d , m } h = m h=m h = m H ( x ∗ , Π , ℓ ) ≤ η ℓ \mathcal{H}(x^*,\Pi,\ell)\leq\eta\ell H ( x ∗ , Π , ℓ ) ≤ η ℓ 0 ≤ ℓ ≤ m 0\leq\ell\leq m 0 ≤ ℓ ≤ m h = n d < m h=nd<m h = n d < m ℓ > h \ell>h ℓ > h n < ℓ / d n<\ell/d n < ℓ / d ℓ \ell ℓ
H ( x ∗ , Π , ℓ ) + ( ℓ − h ) ⋅ τ ( x ∗ , Π ) + I ( x ∗ , Π ) ≤ n d ⋅ h + ( ℓ − h ) ⋅ n d + n < n d ⋅ ℓ + ℓ d = n + 1 d ⋅ ℓ = η ℓ . \begin{aligned}
\mathcal{H}(x^*,\Pi,\ell)+(\ell-h)\cdot\tau(x^*,\Pi)+\mathcal{I}(x^*,\Pi)\leq\frac{n}{d}\cdot h+(\ell-h)\cdot\frac{n}{d}+n \\
&<\frac{n}{d}\cdot\ell+\frac{\ell}{d}=\frac{n+1}{d}\cdot\ell=\eta\ell.
\end{aligned} H ( x ∗ , Π , ℓ ) + ( ℓ − h ) ⋅ τ ( x ∗ , Π ) + I ( x ∗ , Π ) ≤ d n ⋅ h + ( ℓ − h ) ⋅ d n + n < d n ⋅ ℓ + d ℓ = d n + 1 ⋅ ℓ = η ℓ . The lemma follows.
The path forward is now clear: set h = min { n d , m } h=\min\{nd,m\} h = min { n d , m } ( x ∗ , Π ) (x^*,\Pi) ( x ∗ , Π ) Π I = { π i } i ∈ I \Pi_I=\{\pi_i\}_{i\in I} Π I = { π i } i ∈ I I ⊆ N I\subseteq N I ⊆ N ∣ I ∣ ≥ n − O ( log 2 n ) |I|\geq n-O(\log^2 n) ∣ I ∣ ≥ n − O ( log 2 n ) I ( x ∗ , Π ) = n \mathcal{I}(x^*,\Pi)=n I ( x ∗ , Π ) = n O ( log 2 n ) O(\log^2 n) O ( log 2 n )
Theorem 2.2 (Head rounding). Let Π \Pi Π ∣ π ∣ ≤ h |\pi|\leq h ∣ π ∣ ≤ h h ≥ 1 h\geq1 h ≥ 1 i ∈ N i\in N i ∈ N ( x i , Π ) (x_i,\Pi) ( x i , Π ) x = { x i } i ∈ N x=\{x_i\}_{i\in N} x = { x i } i ∈ N y = { y i } i ∈ N y=\{y_i\}_{i\in N} y = { y i } i ∈ N ( y i , Π ) (y_i,\Pi) ( y i , Π ) i , π i,\pi i , π y i ( π ) ∈ { 0 , 1 } y_i(\pi)\in\{0,1\} y i ( π ) ∈ { 0 , 1 } y i ( π ) = 1 y_i(\pi)=1 y i ( π ) = 1 π ≠ ⊥ \pi\ne\perp π = ⊥ x i ( π ) > 0 x_i(\pi)>0 x i ( π ) > 0
H ( y , Π , ℓ ) ≤ H ( x , Π , ℓ ) , ∀ 0 ≤ ℓ ≤ m , \mathcal{H}(y,\Pi,\ell)\leq\mathcal{H}(x,\Pi,\ell),\qquad\forall\,0\leq\ell\leq m, H ( y , Π , ℓ ) ≤ H ( x , Π , ℓ ) , ∀ 0 ≤ ℓ ≤ m , τ ( y , Π ) ≤ τ ( x , Π ) , \tau(y,\Pi)\leq\tau(x,\Pi), τ ( y , Π ) ≤ τ ( x , Π ) , I ( y , Π ) ≤ I ( x , Π ) , \mathcal{I}(y,\Pi)\leq\mathcal{I}(x,\Pi), I ( y , Π ) ≤ I ( x , Π ) , I ( y , Π ) ≥ I ( x , Π ) − ρ ( h , X ) , \mathcal{I}(y,\Pi)\geq\mathcal{I}(x,\Pi)-\rho(h,X), I ( y , Π ) ≥ I ( x , Π ) − ρ ( h , X ) , where X = ∣ { ( i , π ) : 0 < x i ( π ) < 1 } ∣ X=|\{(i,\pi):0<x_i(\pi)<1\}| X = ∣ {( i , π ) : 0 < x i ( π ) < 1 } ∣
ρ ( h , X ) : = 8 + 4 log max { 1 , X } ⋅ ( 1 + ln h + ln max { 1 , X } ) . \rho(h,X):=8+4\log\max\{1,X\}\cdot(1+\ln h+\ln\max\{1,X\}). ρ ( h , X ) := 8 + 4 log max { 1 , X } ⋅ ( 1 + ln h + ln max { 1 , X }) . We prove Theorem 2.2 in Appendix § 9 (Theorem A.1 ), and use it to prove Theorem 2.1.
Proof of Theorem 2.1. Take h = min { n d , m } h=\min\{nd,m\} h = min { n d , m } π i , j \pi_{i,j} π i , j i ∈ N i\in N i ∈ N j ∈ [ d ] j\in[d] j ∈ [ d ] Π : = { ∅ , ⊥ } + ⋃ i ∈ N { π i , j : j ∈ [ d ] } \Pi:=\{\varnothing,\perp\}+\bigcup_{i\in N}\{\pi_{i,j}:j\in[d]\} Π := { ∅ , ⊥ } + ⋃ i ∈ N { π i , j : j ∈ [ d ]} ( x ∗ , Π ) (x^*,\Pi) ( x ∗ , Π ) π i , j \pi_{i,j} π i , j π ∈ Π \pi\in\Pi π ∈ Π ∣ π ∣ ≤ h |\pi|\leq h ∣ π ∣ ≤ h
( 1 − ) (1 - ) ( 1 − ) ( 1 − ) n (1 - )n ( 1 − ) n Note that the total number of strictly fractional entries in x ∗ x^* x ∗ X ∗ ≤ n d X^* \le nd X ∗ ≤ n d ( x ∗ , Π ) (x^*,\Pi) ( x ∗ , Π ) ( y , Π ) (y,\Pi) ( y , Π )
H ( y , Π , ℓ ) ≤ H ( x ∗ , Π , ℓ ) ≤ n d min { ℓ , h } , ∀ 0 ≤ ℓ ≤ m , \mathcal{H}(y,\Pi,\ell) \le\mathcal{H}(x^*,\Pi,\ell) \le\frac{n}{d}\min\{\ell,h\}, \qquad\forall0 \le\ell\le m, H ( y , Π , ℓ ) ≤ H ( x ∗ , Π , ℓ ) ≤ d n min { ℓ , h } , ∀0 ≤ ℓ ≤ m , τ ( y , Π ) ≤ τ ( x ∗ , Π ) ≤ n d \tau(y,\Pi) \le\tau(x^*,\Pi) \le\frac{n}{d} τ ( y , Π ) ≤ τ ( x ∗ , Π ) ≤ d n I ( y , Π ) ≤ I ( x ∗ , Π ) = n , \mathcal{I}(y,\Pi) \le\mathcal{I}(x^*,\Pi)=n, I ( y , Π ) ≤ I ( x ∗ , Π ) = n , I ( y , Π ) ≥ I ( x ∗ , Π ) − ρ ( h , X ∗ ) ≥ n − ρ ( h , n d ) . \mathcal{I}(y,\Pi) \ge\mathcal{I}(x^*,\Pi)-\rho(h,X^*) \ge n-\rho(h,nd). I ( y , Π ) ≥ I ( x ∗ , Π ) − ρ ( h , X ∗ ) ≥ n − ρ ( h , n d ) . Let I ⊆ N I \subseteq N I ⊆ N I ( y , Π ) \mathcal{I}(y,\Pi) I ( y , Π ) i ∈ I i \in I i ∈ I y i ( ⊥ ) = 0 y_i(\bot)=0 y i ( ⊥ ) = 0 π i ∈ Π − ⊥ \pi_i \in\Pi-\bot π i ∈ Π − ⊥ y i ( π i ) = 1 y_i(\pi_i)=1 y i ( π i ) = 1 Π I : = { π i } i ∈ I \Pi_I:=\{\pi_i\}_{i\in I} Π I := { π i } i ∈ I η = ( n + 1 ) / d \eta=(n+1)/d η = ( n + 1 ) / d 0 ≤ ℓ ≤ h 0\le\ell\le h 0 ≤ ℓ ≤ h P ( Π I , ℓ ) = H ( y , Π , ℓ ) ≤ H ( x ∗ , Π , ℓ ) ≤ η ℓ \mathcal{P}(\Pi_I,\ell)=\mathcal{H}(y,\Pi,\ell)\le\mathcal{H}(x^*,\Pi,\ell)\le\eta\ell P ( Π I , ℓ ) = H ( y , Π , ℓ ) ≤ H ( x ∗ , Π , ℓ ) ≤ η ℓ h + 1 ≤ ℓ ≤ m h+1\le\ell\le m h + 1 ≤ ℓ ≤ m
P ( Π I , ℓ ) + ( ℓ − h ) ∑ i ∈ I τ i ( π i ) + ∣ I ∣ = H ( y , Π , ℓ ) + ( ℓ − h ) τ ( y , Π ) + I ( y , Π ) ≤ H ( x ∗ , Π , ℓ ) + ( ℓ − h ) ⋅ τ ( x ∗ , Π ) + I ( x ∗ , Π ) ≤ n d ⋅ h + ( ℓ − h ) ⋅ n d + n < n d ⋅ ℓ + ℓ d = n + 1 d ⋅ ℓ = η ℓ . \begin{aligned}
\mathcal{P}(\Pi_I,\ell)+(\ell-h)\sum_{i\in I}\tau_i(\pi_i)+|I|=\mathcal{H}(y,\Pi,\ell)+(\ell-h)\tau(y,\Pi)+\mathcal{I}(y,\Pi) \\
&\le\mathcal{H}(x^*,\Pi,\ell)+(\ell-h)\cdot\tau(x^*,\Pi)+\mathcal{I}(x^*,\Pi)\le\frac{n}{d}\cdot h+(\ell-h)\cdot\frac{n}{d}+n \\
&<\frac{n}{d}\cdot\ell+\frac{\ell}{d}=\frac{n+1}{d}\cdot\ell=\eta\ell.
\end{aligned} P ( Π I , ℓ ) + ( ℓ − h ) i ∈ I ∑ τ i ( π i ) + ∣ I ∣ = H ( y , Π , ℓ ) + ( ℓ − h ) τ ( y , Π ) + I ( y , Π ) ≤ H ( x ∗ , Π , ℓ ) + ( ℓ − h ) ⋅ τ ( x ∗ , Π ) + I ( x ∗ , Π ) ≤ d n ⋅ h + ( ℓ − h ) ⋅ d n + n < d n ⋅ ℓ + d ℓ = d n + 1 ⋅ ℓ = η ℓ . By construction, for every i ∈ I i\in I i ∈ I T i = 0 T_i=0 T i = 0 Δ ( π i ) = 0 \Delta(\pi_i)=0 Δ ( π i ) = 0 y i ( π i ) = 1 y_i(\pi_i)=1 y i ( π i ) = 1 x i ( π i ) > 0 x_i(\pi_i)>0 x i ( π i ) > 0 π i \pi_i π i i i i Ψ I = { ψ i } i ∈ I \Psi_I=\{\psi_i\}_{i\in I} Ψ I = { ψ i } i ∈ I 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m
P ( Π I , ℓ ) + P ( Ψ I , ℓ ) ≤ η ℓ , \mathcal{P}(\Pi_I,\ell)+\mathcal{P}(\Psi_I,\ell)\le\eta\ell, P ( Π I , ℓ ) + P ( Ψ I , ℓ ) ≤ η ℓ , and for every i ∈ I i\in I i ∈ I v ( π i ) + v ( ψ i ) ≥ 1 v(\pi_i)+v(\psi_i)\ge1 v ( π i ) + v ( ψ i ) ≥ 1 ( I , Π I , Ψ I ) (I,\Pi_I,\Psi_I) ( I , Π I , Ψ I ) F ⊆ M F\subseteq M F ⊆ M { π i , ψ i } i ∈ I \{\pi_i,\psi_i\}_{i\in I} { π i , ψ i } i ∈ I F F F 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m ∣ F ∩ [ ℓ ] ∣ ≤ η ℓ |F\cap[\ell]|\le\eta\ell ∣ F ∩ [ ℓ ] ∣ ≤ η ℓ
Every agent i ∈ I i\in I i ∈ I F i ⊆ F F_i\subseteq F F i ⊆ F v i ( F i ) ≥ v ( π i ) + v ( ψ i ) ≥ 1 v_i(F_i)\ge v(\pi_i)+v(\psi_i)\ge1 v i ( F i ) ≥ v ( π i ) + v ( ψ i ) ≥ 1 U : = N − I U:=N-I U := N − I ∣ I ∣ = I ( y , Π ) ≥ n − ρ ( h , n d ) |I|=\mathcal{I}(y,\Pi)\ge n-\rho(h,nd) ∣ I ∣ = I ( y , Π ) ≥ n − ρ ( h , n d ) ∣ U ∣ ≤ ρ ( h , n d ) |U|\le\rho(h,nd) ∣ U ∣ ≤ ρ ( h , n d ) u ∈ U u\in U u ∈ U v u ( F ) ≤ η d = n + 1 v_u(F)\le\eta d=n+1 v u ( F ) ≤ η d = n + 1 ∣ U ∣ ≤ ( d − n ) / 2 |U|\le(d-n)/2 ∣ U ∣ ≤ ( d − n ) /2 U U U M − F M-F M − F u ∈ U u\in U u ∈ U ∣ U ∣ ≤ ( d − n ) / 2 |U|\le(d-n)/2 ∣ U ∣ ≤ ( d − n ) /2 d d d
ρ ( h , n d ) ≤ ( d − n ) / 2 ⟺ d ≥ n + 2 ρ ( h , n d ) . \rho(h,nd)\le(d-n)/2 \Longleftrightarrow d\ge n+2\rho(h,nd). ρ ( h , n d ) ≤ ( d − n ) /2 ⟺ d ≥ n + 2 ρ ( h , n d ) . We now estimate ρ ( h , n d ) \rho(h,nd) ρ ( h , n d ) h , X ∗ ≤ n d h,X^*\le nd h , X ∗ ≤ n d d ≤ 2 n − 1 d\le2n-1 d ≤ 2 n − 1 d > 2 n − 1 d>2n-1 d > 2 n − 1
ρ ( h , n d ) = 8 + 4 log ( n d ) ⋅ ( 1 + ln ( n d ) + ln ( n d ) ) \rho(h,nd)=8+4\log(nd)\cdot(1+\ln(nd)+\ln(nd)) ρ ( h , n d ) = 8 + 4 log ( n d ) ⋅ ( 1 + ln ( n d ) + ln ( n d )) ≤ 8 + 4 log ( n d ) + 8 ln 2 log 2 ( n d ) ≤ 8 + 4 log ( 2 n 2 ) + 8 ln 2 log 2 ( 2 n 2 ) . \le8+4\log(nd)+8\ln2\log^2(nd)\le8+4\log(2n^2)+8\ln2\log^2(2n^2). ≤ 8 + 4 log ( n d ) + 8 ln 2 log 2 ( n d ) ≤ 8 + 4 log ( 2 n 2 ) + 8 ln 2 log 2 ( 2 n 2 ) . Setting z = 1 + log n z=1+\log n z = 1 + log n
ρ ( h , n d ) ≤ 8 + 4 ( 2 z − 1 ) + 8 ln 2 ( 2 z − 1 ) 2 = 32 ln 2 z 2 + ( 8 − 32 ln 2 ) z + 4 + 8 ln 2 < 32 ln 2 z 2 < 23 ( 1 + log n ) 2 . \begin{aligned}
\rho(h,nd)\le8+4(2z-1)+8\ln2(2z-1)^2 \\
&=32\ln2z^2+(8-32\ln2)z+4+8\ln2<32\ln2z^2<23(1+\log n)^2.
\end{aligned} ρ ( h , n d ) ≤ 8 + 4 ( 2 z − 1 ) + 8 ln 2 ( 2 z − 1 ) 2 = 32 ln 2 z 2 + ( 8 − 32 ln 2 ) z + 4 + 8 ln 2 < 32 ln 2 z 2 < 23 ( 1 + log n ) 2 . As a result, taking d ≥ n + 46 ( 1 + log n ) 2 d\ge n+46(1+\log n)^2 d ≥ n + 46 ( 1 + log n ) 2
( 1 − δ ) (1-\delta) ( 1 − δ ) ( 1 − ε ) n (1-\varepsilon)n ( 1 − ε ) n In this section, we prove Theorem 1.1. We assume m ≥ n m \ge n m ≥ n
Theorem 3.1 (Theorem 1.1 restated). For every fixed ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 ) q ≥ 3 q \ge3 q ≥ 3 n ( ε , q ) n(\varepsilon,q) n ( ε , q ) n ≥ n ( ε , q ) n \ge n(\varepsilon,q) n ≥ n ( ε , q )
Consider a fair allocation instance with m m m M M M n n n N N N i ∈ N i \in N i ∈ N v i v_i v i i ∈ N i \in N i ∈ N MMS ( M , v i , d ) = 1 \operatorname{MMS}(M,v_i,d)=1 MMS ( M , v i , d ) = 1 1 ≤ d ≤ ( 1 − ε ) n 1 \le d \le(1-\varepsilon)n 1 ≤ d ≤ ( 1 − ε ) n I = { i 1 , … , i d } ⊆ N I=\{i_1,\ldots,i_d\} \subseteq N I = { i 1 , … , i d } ⊆ N ∣ I ∣ = d |I|=d ∣ I ∣ = d ( B i 1 , … , B i d ) (B_{i_1},\ldots,B_{i_d}) ( B i 1 , … , B i d ) M M M I I I i ∈ I i \in I i ∈ I v i ( B i ) ≥ 1 − 2 / q v_i(B_i) \ge1-2/q v i ( B i ) ≥ 1 − 2/ q
The core of our approach will be essentially the same as in the proof of Theorem 2.1. That is, we would like to find a collection of profiles Π I = { π i } i ∈ I \Pi_I=\{\pi_i\}_{i\in I} Π I = { π i } i ∈ I I ⊆ N I \subseteq N I ⊆ N ∣ I ∣ ≥ d |I| \ge d ∣ I ∣ ≥ d i ∈ I i \in I i ∈ I v ( π i ) ≥ 1 − 2 / q v(\pi_i) \ge1-2/q v ( π i ) ≥ 1 − 2/ q 0 ≤ ℓ ≤ m 0 \le\ell\le m 0 ≤ ℓ ≤ m
P ( Π I , ℓ ) ≤ ℓ . \mathcal{P}(\Pi_I,\ell) \le\ell. P ( Π I , ℓ ) ≤ ℓ . Then, we obtain the desired ( 1 − 2 / q ) (1-2/q) ( 1 − 2/ q ) I , Π I I,\Pi_I I , Π I
Head and tail profiles with different thresholds
As before, we will first find a collection Π I \Pi_I Π I π i ∈ Π I \pi_i \in\Pi_I π i ∈ Π I ψ i ∈ Ψ I \psi_i \in\Psi_I ψ i ∈ Ψ I i ∈ I i \in I i ∈ I v ( π i ) + ψ ( π i ) ≥ 1 − 2 / q v(\pi_i)+\psi(\pi_i) \ge1-2/q v ( π i ) + ψ ( π i ) ≥ 1 − 2/ q i i i { π i , ψ i } \{\pi_i,\psi_i\} { π i , ψ i } i ∈ N i \in N i ∈ N head threshold h i h_i h i
Definition 3.1. For an integer q > 0 q>0 q > 0 M = [ m ] M=[m] M = [ m ] i ∈ N i \in N i ∈ N
h i : = ∣ { e ∈ M : v i ( e ) ≥ 1 / q } ∣ . h_i:=\lvert\{e \in M:v_i(e)\ge1/q\}\rvert. h i := ∣{ e ∈ M : v i ( e ) ≥ 1/ q }∣ . We refer to items { 1 , … , h i } \{1,\ldots,h_i\} { 1 , … , h i } head items of agent i i i { h i + 1 , … , m } \{h_i+1,\ldots,m\} { h i + 1 , … , m } tail items . For agent i ∈ N i \in N i ∈ N tail value of i i i
T i = ∑ e > h i v i ( e ) . T_i=\sum_{e>h_i}v_i(e). T i = e > h i ∑ v i ( e ) . In the proof of Theorem 2.1, it was of no difference for us how many different head profiles there are for agents i ∈ N i \in N i ∈ N ( 0 , 1 ] (0,1] ( 0 , 1 ] n d = O ( n 2 ) n^d=O(n^2) n d = O ( n 2 ) 1 / q 2 1/q^2 1/ q 2
Definition 3.2. For an integer q > 0 q>0 q > 0 head demand is a positive value α ∈ ( 0 , 1 ] \alpha\in(0,1] α ∈ ( 0 , 1 ]
α = q + k q 2 \alpha=\frac{q+k}{q^2} α = q 2 q + k for integer k ∈ { 0 , 1 , 2 , … , q 2 − q } k \in\{0,1,2,\ldots,q^2-q\} k ∈ { 0 , 1 , 2 , … , q 2 − q } α \alpha α
1 q , 1 q + 1 q 2 , 1 q + 2 q 2 , … , 1 q + k q 2 , … , 1. \frac{1}{q},\ \frac{1}{q}+\frac{1}{q^2},\ \frac{1}{q}+\frac{2}{q^2},\ \ldots,\ \frac{1}{q}+\frac{k}{q^2},\ \ldots,\ 1. q 1 , q 1 + q 2 1 , q 1 + q 2 2 , … , q 1 + q 2 k , … , 1. A head profile is a finite multiset π = { α 1 , … , α ∣ π ∣ } \pi=\{\alpha_1,\ldots,\alpha_{|\pi|}\} π = { α 1 , … , α ∣ π ∣ } tail demand is a positive value β ∈ ( 0 , 1 ] \beta\in(0,1] β ∈ ( 0 , 1 ] β ≤ 1 / q \beta\le1/q β ≤ 1/ q β = 1 / q \beta=1/q β = 1/ q tail profile is a finite multiset ψ = { β 1 , … , β ∣ ψ ∣ } \psi=\{\beta_1,\ldots,\beta_{|\psi|}\} ψ = { β 1 , … , β ∣ ψ ∣ }
Note that for every agent i ∈ N i \in N i ∈ N B ⊆ M B \subseteq M B ⊆ M v i ( B ) = 1 v_i(B)=1 v i ( B ) = 1 q q q B B B 1 / q 2 1/q^2 1/ q 2 1 − q ⋅ ( 1 / q 2 ) = 1 − 1 / q 1-q\cdot(1/q^2)=1-1/q 1 − q ⋅ ( 1/ q 2 ) = 1 − 1/ q 1 / q 1/q 1/ q 1 − 1 / q 1-1/q 1 − 1/ q 1 1 1
Definition 3.3. A deficit of a head profile π \pi π
Δ ( π ) : = ( 1 − 1 / q − v ( π ) ) + . \Delta(\pi) := (1-1/q-v(\pi))_+. Δ ( π ) := ( 1 − 1/ q − v ( π ) ) + . As in the proof of Theorem 2.1, if an agent i ∈ N i \in N i ∈ N π i ≠ ⊥ \pi_i \ne\bot π i = ⊥ Δ ( π i ) > 0 \Delta(\pi_i)>0 Δ ( π i ) > 0 π i \pi_i π i ψ i \psi_i ψ i
There are two key differences from the original Lemma 2.3. First, for every agent i ∈ I i \in I i ∈ I h i h_i h i ( ℓ − h ) (\ell-h) ( ℓ − h ) ( ℓ − h i ) (\ell-h_i) ( ℓ − h i ) ∣ I ∣ |I| ∣ I ∣ [5 , 12 ] . We achieve this by taking the last item in each segment of length 1 / τ i ( π i ) 1/\tau_i(\pi_i) 1/ τ i ( π i ) I I I 1 / q 1/q 1/ q
Lemma 3.1 (Tail completion). Let I ⊆ N I \subseteq N I ⊆ N π i \pi_i π i i ∈ I i \in I i ∈ I Π I = { π i } i ∈ I \Pi_I=\{\pi_i\}_{i\in I} Π I = { π i } i ∈ I T i = 0 T_i=0 T i = 0 Δ ( π i ) = 0 \Delta(\pi_i)=0 Δ ( π i ) = 0 i ∈ I i \in I i ∈ I 0 ≤ ℓ ≤ m 0 \leq\ell\leq m 0 ≤ ℓ ≤ m
P ( Π I , ℓ ) ≤ ℓ − ∑ i ∈ I τ i ( π i ) ( ℓ − h i ) + , \mathcal{P}(\Pi_I,\ell) \leq\ell-\sum_{i\in I}\tau_i(\pi_i)(\ell-h_i)_+, P ( Π I , ℓ ) ≤ ℓ − i ∈ I ∑ τ i ( π i ) ( ℓ − h i ) + , there exists a collection Ψ I : = { ψ i } i ∈ I \Psi_I:=\{\psi_i\}_{i\in I} Ψ I := { ψ i } i ∈ I i i i ψ i \psi_i ψ i
∀ 0 ≤ ℓ ≤ m : P ( Π I , ℓ ) + P ( Ψ I , ℓ ) ≤ ℓ \forall0 \leq\ell\leq m:\quad\mathcal{P}(\Pi_I,\ell)+\mathcal{P}(\Psi_I,\ell)\leq\ell ∀0 ≤ ℓ ≤ m : P ( Π I , ℓ ) + P ( Ψ I , ℓ ) ≤ ℓ and v ( π i ) + v ( ψ i ) ≥ 1 − 2 / q v(\pi_i)+v(\psi_i)\geq1-2/q v ( π i ) + v ( ψ i ) ≥ 1 − 2/ q i ∈ I i \in I i ∈ I
Proof. Consider i ∈ I i \in I i ∈ I τ i : = τ i ( π i ) \tau_i:=\tau_i(\pi_i) τ i := τ i ( π i ) τ i = 0 \tau_i=0 τ i = 0 ψ i = ∅ \psi_i=\varnothing ψ i = ∅ ψ i \psi_i ψ i i i i { h i + 1 , … , m } \{h_i+1,\ldots,m\} { h i + 1 , … , m } 1 / τ i 1/\tau_i 1/ τ i ψ i \psi_i ψ i t = 1 , … , ⌊ τ i ⋅ ( m − h i ) ⌋ t=1,\ldots,\lfloor\tau_i\cdot(m-h_i)\rfloor t = 1 , … , ⌊ τ i ⋅ ( m − h i )⌋ v i ( e i , t ) v_i(e_{i,t}) v i ( e i , t ) ψ i \psi_i ψ i
e i , t = h i + ⌈ t τ i ⌉ . e_{i,t}=h_i+\left\lceil\frac{t}{\tau_i}\right\rceil. e i , t = h i + ⌈ τ i t ⌉ . Note that in this construction, for every item we take, its value is at least as large as if we took a τ i \tau_i τ i 1 / τ i 1/\tau_i 1/ τ i immediately after the taken item. So, the total value of selected items is at least as large as the τ i \tau_i τ i T i T_i T i 1 / q 1/q 1/ q
We will now prove this formally. Similar to Lemma 2.3, the construction of ψ i \psi_i ψ i 1 ≤ k ≤ m − h i 1\leq k\leq m-h_i 1 ≤ k ≤ m − h i β i , k : = v i ( h i + k ) \beta_{i,k}:=v_i(h_i+k) β i , k := v i ( h i + k ) k ∈ [ m − h i ] k\in[m-h_i] k ∈ [ m − h i ]
⌊ τ i ⋅ k ⌋ − ⌊ τ i ⋅ ( k − 1 ) ⌋ \left\lfloor\tau_i\cdot k\right\rfloor-\left\lfloor\tau_i\cdot(k-1)\right\rfloor ⌊ τ i ⋅ k ⌋ − ⌊ τ i ⋅ ( k − 1 ) ⌋ copies of β i , k \beta_{i,k} β i , k ψ i \psi_i ψ i
e i , t ≤ h i + k ⟺ ⌈ t τ i ⌉ ≤ k ⟺ t ≤ τ i ⋅ k ⟺ t ≤ ⌊ τ i ⋅ k ⌋ . e_{i,t}\leq h_i+k\Longleftrightarrow\left\lceil\frac{t}{\tau_i}\right\rceil\leq k\Longleftrightarrow t\leq\tau_i\cdot k\Longleftrightarrow t\leq\left\lfloor\tau_i\cdot k\right\rfloor. e i , t ≤ h i + k ⟺ ⌈ τ i t ⌉ ≤ k ⟺ t ≤ τ i ⋅ k ⟺ t ≤ ⌊ τ i ⋅ k ⌋ . So, the first k k k ⌊ τ i ⋅ k ⌋ \lfloor\tau_i \cdot k \rfloor ⌊ τ i ⋅ k ⌋ ψ i \psi_i ψ i k k k ⌊ τ i ⋅ k ⌋ − ⌊ τ i ⋅ ( k − 1 ) ⌋ \lfloor\tau_i \cdot k \rfloor- \lfloor\tau_i \cdot(k - 1) \rfloor ⌊ τ i ⋅ k ⌋ − ⌊ τ i ⋅ ( k − 1 )⌋ τ i > 1 \tau_i > 1 τ i > 1 i i i ψ i \psi_i ψ i i ∈ I i \in I i ∈ I τ i > 1 \tau_i > 1 τ i > 1 i ∈ I i \in I i ∈ I π i = ∅ \pi_i = \varnothing π i = ∅ T i > 0 T_i > 0 T i > 0 π i = ∅ \pi_i = \varnothing π i = ∅ ψ i \psi_i ψ i
A straightforward computation shows that our choice of ψ i \psi_i ψ i i i i
v ( ψ i ) = ∑ k = 1 m − h i β i , k ⋅ ( ⌊ τ i ⋅ k ⌋ − ⌊ τ i ⋅ ( k − 1 ) ⌋ ) = ∑ k = 1 m − h i ⌊ τ i ⋅ k ⌋ ⋅ ( β i , k − β i , k + 1 ) ≥ ∑ k = 1 m − h i ( τ i ⋅ k − 1 ) ⋅ ( β i , k − β i , k + 1 ) = τ i ∑ k = 1 m − h i β i , k − β i , 1 = Δ ( π i ) − β i , 1 ≥ Δ ( π i ) − 1 / q . \begin{aligned}
v(\psi_i) = \sum_{k=1}^{m-h_i} \beta_{i,k} \cdot\left(\left\lfloor\tau_i \cdot k \right\rfloor- \left\lfloor\tau_i \cdot(k - 1) \right\rfloor\right) = \sum_{k=1}^{m-h_i} \left\lfloor\tau_i \cdot k \right\rfloor\cdot(\beta_{i,k} - \beta_{i,k+1}) \\
&\ge\sum_{k=1}^{m-h_i} (\tau_i \cdot k - 1) \cdot(\beta_{i,k} - \beta_{i,k+1}) = \tau_i \sum_{k=1}^{m-h_i} \beta_{i,k} - \beta_{i,1} = \Delta(\pi_i) - \beta_{i,1} \ge\Delta(\pi_i) - 1/q.
\end{aligned} v ( ψ i ) = k = 1 ∑ m − h i β i , k ⋅ ( ⌊ τ i ⋅ k ⌋ − ⌊ τ i ⋅ ( k − 1 ) ⌋ ) = k = 1 ∑ m − h i ⌊ τ i ⋅ k ⌋ ⋅ ( β i , k − β i , k + 1 ) ≥ k = 1 ∑ m − h i ( τ i ⋅ k − 1 ) ⋅ ( β i , k − β i , k + 1 ) = τ i k = 1 ∑ m − h i β i , k − β i , 1 = Δ ( π i ) − β i , 1 ≥ Δ ( π i ) − 1/ q . Since Δ ( π i ) = ( 1 − 1 / q − v ( π i ) ) + \Delta(\pi_i) = (1 - 1/q - v(\pi_i))^+ Δ ( π i ) = ( 1 − 1/ q − v ( π i ) ) + v ( π i ) + v ( ψ i ) ≥ 1 − 2 / q v(\pi_i) + v(\psi_i) \ge1 - 2/q v ( π i ) + v ( ψ i ) ≥ 1 − 2/ q
We will now bound the additional prefix load generated by Ψ I = { ψ i } i ∈ I \Psi_I = \{\psi_i\}_{i \in I} Ψ I = { ψ i } i ∈ I 0 ≤ ℓ ≤ h i 0 \le\ell\le h_i 0 ≤ ℓ ≤ h i P i ( ψ i , ℓ ) = 0 \mathcal{P}_i(\psi_i,\ell) = 0 P i ( ψ i , ℓ ) = 0 i ∈ I i \in I i ∈ I ℓ > h i \ell> h_i ℓ > h i ψ i \psi_i ψ i k = ℓ − h i k = \ell- h_i k = ℓ − h i ⌊ τ i ⋅ k ⌋ ≤ τ i ⋅ k \lfloor\tau_i \cdot k \rfloor\le\tau_i \cdot k ⌊ τ i ⋅ k ⌋ ≤ τ i ⋅ k Ψ I \Psi_I Ψ I ℓ \ell ℓ
P ( Ψ I , ℓ ) = ∑ i ∈ I P i ( ψ i , ℓ ) ≤ ∑ i ∈ I τ i ⋅ ( ℓ − h i ) + . \mathcal{P}(\Psi_I,\ell) = \sum_{i \in I} \mathcal{P}_i(\psi_i,\ell) \le\sum_{i \in I} \tau_i \cdot(\ell- h_i)_+. P ( Ψ I , ℓ ) = i ∈ I ∑ P i ( ψ i , ℓ ) ≤ i ∈ I ∑ τ i ⋅ ( ℓ − h i ) + . The lemma follows.
Note that the main difference of Lemma 3.1 from the usual faithful rounding setting is that every agent i ∈ I i \in I i ∈ I I I I
Fractional head profiles and the large tail case
We now focus on finding a collection of head profiles Π I \Pi_I Π I I ⊆ N I \subseteq N I ⊆ N i ∈ N i \in N i ∈ N 1 / q 2 1/q^2 1/ q 2
Definition 3.4. For every agent i ∈ N i \in N i ∈ N B i = { B i , 1 , … , B i , d } \mathcal{B}_i = \{B_{i,1}, \ldots, B_{i,d}\} B i = { B i , 1 , … , B i , d } d d d M M M MMS ( M , v i , d ) \operatorname{MMS}(M,v_i,d) MMS ( M , v i , d ) B i , j ∈ B i B_{i,j} \in\mathcal{B}_i B i , j ∈ B i v i ( B i , j ) = 1 v_i(B_{i,j}) = 1 v i ( B i , j ) = 1 j ∈ [ d ] j \in[d] j ∈ [ d ]
π i , j : = { ⌊ q 2 ⋅ v i ( e ) ⌋ q 2 : e ∈ B i , j , v i ( e ) ≥ 1 / q } . \pi_{i,j} := \left\{\frac{\left\lfloor q^2 \cdot v_i(e) \right\rfloor}{q^2} : e \in B_{i,j},\ v_i(e) \ge1/q\right\}. π i , j := { q 2 ⌊ q 2 ⋅ v i ( e ) ⌋ : e ∈ B i , j , v i ( e ) ≥ 1/ q } . Let Π : = { ∅ , ⊥ } + ⋃ i ∈ N { π i , j : j ∈ [ d ] } \Pi:= \{\varnothing,\bot\} + \bigcup_{i \in N} \{\pi_{i,j} : j \in[d]\} Π := { ∅ , ⊥ } + ⋃ i ∈ N { π i , j : j ∈ [ d ]}
As in the proof of Theorem 2.1, we will move to fractional head profiles and use rounding to get an integral collection. All the definitions of fractional head profiles ( x , Π ) (x,\Pi) ( x , Π ) ℓ \ell ℓ H ( x , Π , ℓ ) \mathcal{H}(x,\Pi,\ell) H ( x , Π , ℓ ) I ( x , Π ) \mathcal{I}(x,\Pi) I ( x , Π ) § 2.4 . However, due to the fact that instead of a common head threshold h h h h i h_i h i i ∈ N i \in N i ∈ N 0 ≤ ℓ ≤ m 0 \le\ell\le m 0 ≤ ℓ ≤ m
Definition 3.5. For agent i ∈ N i \in N i ∈ N fractional head profile ( x i , Π ) (x_i,\Pi) ( x i , Π ) fractional ℓ \ell ℓ Π \Pi Π is
T i ( x i , Π , ℓ ) : = ∑ π ∈ Π x i ( π ) ⋅ τ i ( π ) ( ℓ − h i ) + = τ i ( x i , Π ) ⋅ ( ℓ − h i ) + . \mathcal{T}_i(x_i,\Pi,\ell) := \sum_{\pi\in\Pi} x_i(\pi)\cdot\tau_i(\pi)(\ell-h_i)_+ = \tau_i(x_i,\Pi)\cdot(\ell-h_i)_+. T i ( x i , Π , ℓ ) := π ∈ Π ∑ x i ( π ) ⋅ τ i ( π ) ( ℓ − h i ) + = τ i ( x i , Π ) ⋅ ( ℓ − h i ) + . Assume that every agent i ∈ N i \in N i ∈ N fractional head profile ( x i , Π ) (x_i,\Pi) ( x i , Π ) Π \Pi Π x = { x i } i ∈ N x=\{x_i\}_{i\in N} x = { x i } i ∈ N 0 ≤ ℓ ≤ m 0\leq\ell\leq m 0 ≤ ℓ ≤ m total fractional ℓ \ell ℓ Π \Pi Π x x x is
T ( x , Π , ℓ ) : = ∑ i ∈ N T i ( x i , Π , ℓ ) = ∑ i ∈ N τ i ( x i , Π ) ⋅ ( ℓ − h i ) + . \mathcal{T}(x,\Pi,\ell) := \sum_{i\in N}\mathcal{T}_i(x_i,\Pi,\ell)=\sum_{i\in N}\tau_i(x_i,\Pi)\cdot(\ell-h_i)_+. T ( x , Π , ℓ ) := i ∈ N ∑ T i ( x i , Π , ℓ ) = i ∈ N ∑ τ i ( x i , Π ) ⋅ ( ℓ − h i ) + . We define the fractional MMS head profiles a bit differently from § 2.4 : instead of taking a uniform distribution over π i , j \pi_{i,j} π i , j i ∈ N i\in N i ∈ N π i , j \pi_{i,j} π i , j 1 / n 1/n 1/ n 1 − d / n 1-d/n 1 − d / n ⊥ \perp ⊥
Definition 3.6. Let Π : = { ∅ , ⊥ } + ⋃ i ∈ N { π i , j : j ∈ [ d ] } \Pi:=\{\varnothing,\perp\}+\bigcup_{i\in N}\{\pi_{i,j}:j\in[d]\} Π := { ∅ , ⊥ } + ⋃ i ∈ N { π i , j : j ∈ [ d ]} i ∈ N i\in N i ∈ N ( x i ∗ , Π ) (x_i^*,\Pi) ( x i ∗ , Π ) x i ( ⊥ ) = 1 − d / n x_i(\perp)=1-d/n x i ( ⊥ ) = 1 − d / n π ∈ Π − ⊥ \pi\in\Pi-\perp π ∈ Π − ⊥
x i ∗ ( π ) = ∣ { j ∈ [ d ] : π i , j = π } ∣ n . x_i^*(\pi)=\frac{|\{j\in[d]:\pi_{i,j}=\pi\}|}{n}. x i ∗ ( π ) = n ∣ { j ∈ [ d ] : π i , j = π } ∣ . We will refer to ( x i ∗ , Π ) (x_i^*,\Pi) ( x i ∗ , Π ) fractional MMS head profile of agent i i i
We prove an analogue of Lemma 2.4, showing that ( x ∗ , Π ) (x^*,\Pi) ( x ∗ , Π )
Lemma 3.2. Consider the fractional MMS head profiles ( x ∗ , Π ) (x^*,\Pi) ( x ∗ , Π ) i ∈ N i\in N i ∈ N
I ( x ∗ , Π ) = d \mathcal{I}(x^*,\Pi)=d I ( x ∗ , Π ) = d
for every 0 ≤ ℓ ≤ m 0\leq\ell\leq m 0 ≤ ℓ ≤ m
H ( x ∗ , Π , ℓ ) ≤ ℓ − 1 n ∑ i ∈ N ( ℓ − h i ) + ; \mathcal{H}(x^*,\Pi,\ell)\leq\ell-\frac{1}{n}\sum_{i\in N}(\ell-h_i)_+; H ( x ∗ , Π , ℓ ) ≤ ℓ − n 1 i ∈ N ∑ ( ℓ − h i ) + ; for every i ∈ N i\in N i ∈ N
∑ j = 1 d Δ ( π i , j ) ≤ ( 1 − 1 / q ) T i ; \sum_{j=1}^{d}\Delta(\pi_{i,j})\leq(1-1/q)T_i; j = 1 ∑ d Δ ( π i , j ) ≤ ( 1 − 1/ q ) T i ; The ( 1 − 1 / q ) (1-1/q) ( 1 − 1/ q ) 1 / q 2 1/q^2 1/ q 2 1 − 1 / q 1-1/q 1 − 1/ q
Proof. The identity I ( x ∗ , Π ) = d \mathcal{I}(x^*,\Pi)=d I ( x ∗ , Π ) = d ⊥ \perp ⊥ d / n d/n d / n i ∈ N i\in N i ∈ N 0 ≤ ℓ ≤ m 0\leq\ell\leq m 0 ≤ ℓ ≤ m
∑ j = 1 d P i ( π i , j , ℓ ) ≤ min { ℓ , h i } , \sum_{j=1}^{d}\mathcal{P}_i(\pi_{i,j},\ell)\leq\min\{\ell,h_i\}, j = 1 ∑ d P i ( π i , j , ℓ ) ≤ min { ℓ , h i } , which immediately gives by definition of H ( x ∗ , Π , ℓ ) = 1 n ∑ i ∈ N ∑ j = 1 d P i ( π i , j , ℓ ) \mathcal{H}(x^*,\Pi,\ell)=\frac{1}{n}\sum_{i\in N}\sum_{j=1}^{d}\mathcal{P}_i(\pi_{i,j},\ell) H ( x ∗ , Π , ℓ ) = n 1 ∑ i ∈ N ∑ j = 1 d P i ( π i , j , ℓ )
1 n ∑ i ∈ N ∑ j = 1 d P i ( π i , j , ℓ ) + 1 n ∑ i ∈ N ( ℓ − h i ) + ≤ 1 n ∑ i ∈ N ( min { ℓ , h i } + ( ℓ − h i ) + ) = ℓ . \frac{1}{n}\sum_{i\in N}\sum_{j=1}^{d}\mathcal{P}_i(\pi_{i,j},\ell)+\frac{1}{n}\sum_{i\in N}(\ell-h_i)_+
\leq\frac{1}{n}\sum_{i\in N}\left(\min\{\ell,h_i\}+(\ell-h_i)_+\right)=\ell. n 1 i ∈ N ∑ j = 1 ∑ d P i ( π i , j , ℓ ) + n 1 i ∈ N ∑ ( ℓ − h i ) + ≤ n 1 i ∈ N ∑ ( min { ℓ , h i } + ( ℓ − h i ) + ) = ℓ . Now, fix agent i ∈ N i \in N i ∈ N j ∈ [ d ] j \in[d] j ∈ [ d ] v ( B i , j ) = 1 v(B_{i,j}) = 1 v ( B i , j ) = 1 B i , j B_{i,j} B i , j 1 − T i , j 1 - T_{i,j} 1 − T i , j
T i , j : = ∑ e ∈ B i , j e > h i v i ( e ) . T_{i,j} := \sum_{\substack{e \in B_{i,j} \\ e > h_i}} v_i(e). T i , j := e ∈ B i , j e > h i ∑ v i ( e ) . Since each demand in π i , j \pi_{i,j} π i , j 1 / q 1/q 1/ q
∣ π i , j ∣ q ≤ v ( π i , j ) ≤ 1 − T i , j . \frac{|\pi_{i,j}|}{q} \le v(\pi_{i,j}) \le1 - T_{i,j}. q ∣ π i , j ∣ ≤ v ( π i , j ) ≤ 1 − T i , j . For every head item e ∈ [ h i ] e \in[h_i] e ∈ [ h i ]
v ^ i ( e ) : = ⌊ q 2 ⋅ v i ( e ) ⌋ q 2 . \hat{v}_i(e) := \frac{\left\lfloor q^2 \cdot v_i(e) \right\rfloor}{q^2}. v ^ i ( e ) := q 2 ⌊ q 2 ⋅ v i ( e ) ⌋ . We round each v i ( e ) v_i(e) v i ( e ) e ≤ h i e \le h_i e ≤ h i 1 / q 2 1/q^2 1/ q 2 0 ≤ v i ( e ) − v ^ i ( e ) ≤ 1 / q 2 0 \le v_i(e) - \hat{v}_i(e) \le1/q^2 0 ≤ v i ( e ) − v ^ i ( e ) ≤ 1/ q 2 π i , j \pi_{i,j} π i , j
ρ i , j : = ∑ e ∈ B i , j e ≤ h i ( v i ( e ) − v ^ i ( e ) ) ≤ ∣ π i , j ∣ q 2 ≤ 1 − T i , j q . \rho_{i,j} := \sum_{\substack{e \in B_{i,j} \\ e \le h_i}} \left(v_i(e) - \hat{v}_i(e)\right) \le\frac{|\pi_{i,j}|}{q^2} \le\frac{1 - T_{i,j}}{q}. ρ i , j := e ∈ B i , j e ≤ h i ∑ ( v i ( e ) − v ^ i ( e ) ) ≤ q 2 ∣ π i , j ∣ ≤ q 1 − T i , j . At the same time,
v ( π i , j ) = v ^ i ( B i , j ) = ∑ e ∈ B i , j e ≤ h i v ^ i ( e ) = ∑ e ∈ B i , j e ≤ h i v i ( e ) − ρ i , j = 1 − T i , j − ρ i , j . v(\pi_{i,j}) = \hat{v}_i(B_{i,j}) = \sum_{\substack{e \in B_{i,j} \\ e \le h_i}} \hat{v}_i(e) = \sum_{\substack{e \in B_{i,j} \\ e \le h_i}} v_i(e) - \rho_{i,j} = 1 - T_{i,j} - \rho_{i,j}. v ( π i , j ) = v ^ i ( B i , j ) = e ∈ B i , j e ≤ h i ∑ v ^ i ( e ) = e ∈ B i , j e ≤ h i ∑ v i ( e ) − ρ i , j = 1 − T i , j − ρ i , j . As a result, the deficit of π i , j \pi_{i,j} π i , j
Δ ( π i , j ) = ( 1 − 1 / q − v ( π i , j ) ) + = ( T i , j + ρ i , j − 1 / q ) + ≤ ( T i , j + 1 − T i , j q − 1 q ) + = ( 1 − 1 / q ) T i , j \Delta(\pi_{i,j}) = (1 - 1/q - v(\pi_{i,j}))_+ = (T_{i,j} + \rho_{i,j} - 1/q)_+ \le\left(T_{i,j} + \frac{1 - T_{i,j}}{q} - \frac{1}{q}\right)_+ = (1 - 1/q)T_{i,j} Δ ( π i , j ) = ( 1 − 1/ q − v ( π i , j ) ) + = ( T i , j + ρ i , j − 1/ q ) + ≤ ( T i , j + q 1 − T i , j − q 1 ) + = ( 1 − 1/ q ) T i , j and the total deficit of MMS profiles of agent i ∈ N i \in N i ∈ N
∑ j = 1 d Δ ( π i , j ) ≤ ( 1 − 1 / q ) ∑ j = 1 d T i , j = ( 1 − 1 / q ) T i . \sum_{j=1}^{d} \Delta(\pi_{i,j}) \le(1 - 1/q)\sum_{j=1}^{d} T_{i,j} = (1 - 1/q)T_i. j = 1 ∑ d Δ ( π i , j ) ≤ ( 1 − 1/ q ) j = 1 ∑ d T i , j = ( 1 − 1/ q ) T i . It immediately follows that
T ( x ∗ , Π , ℓ ) = 1 n ∑ i ∈ N ∑ j = 1 d τ i ( π i , j ) ( ℓ − h i ) + = 1 n ∑ i ∈ N T i > 0 ( ℓ − h i ) + T i ∑ j = 1 d Δ ( π i , j ) ≤ 1 − 1 / q n ∑ i ∈ N ( ℓ − h i ) + . \mathcal{T}(x^*,\Pi,\ell) = \frac{1}{n}\sum_{i \in N}\sum_{j=1}^{d} \tau_i(\pi_{i,j})(\ell- h_i)_+ = \frac{1}{n}\sum_{\substack{i \in N \\ T_i > 0}} \frac{(\ell- h_i)_+}{T_i}\sum_{j=1}^{d}\Delta(\pi_{i,j}) \le\frac{1 - 1/q}{n}\sum_{i \in N}(\ell- h_i)_+. T ( x ∗ , Π , ℓ ) = n 1 i ∈ N ∑ j = 1 ∑ d τ i ( π i , j ) ( ℓ − h i ) + = n 1 i ∈ N T i > 0 ∑ T i ( ℓ − h i ) + j = 1 ∑ d Δ ( π i , j ) ≤ n 1 − 1/ q i ∈ N ∑ ( ℓ − h i ) + . □ \square □
As in § 2.4 , we will round the fractional MMS head profiles ( x ∗ , Π ) (x^*,\Pi) ( x ∗ , Π ) Π I = { π i } i ∈ I \Pi_I = \{\pi_i\}_{i \in I} Π I = { π i } i ∈ I π ∈ Π \pi\in\Pi π ∈ Π q q q m + 1 m + 1 m + 1 λ > 1 \lambda> 1 λ > 1 I ( x ∗ , Π ) − O q ( log n ) \mathcal{I}(x^*,\Pi) - O_q(\log n) I ( x ∗ , Π ) − O q ( log n ) ( 1 − 1 / q ) (1 - 1/q) ( 1 − 1/ q ) ( x ∗ , Π ) (x^*,\Pi) ( x ∗ , Π ) λ \lambda λ
Theorem 3.2 (Head and tail rounding). Let Π \Pi Π ∣ π ∣ ≤ q |\pi| \le q ∣ π ∣ ≤ q q ≥ 3 q \ge3 q ≥ 3 i ∈ N i \in\mathbb{N} i ∈ N ( x i , Π ) (x_i,\Pi) ( x i , Π ) x = { x i } i ∈ N x = \{x_i\}_{i\in\mathbb{N}} x = { x i } i ∈ N π ∈ Π − ⊥ \pi\in\Pi- \bot π ∈ Π − ⊥ Δ ( π ) = 0 \Delta(\pi) = 0 Δ ( π ) = 0 Δ ( π ) ≥ δ > 0 \Delta(\pi) \ge\delta> 0 Δ ( π ) ≥ δ > 0 δ ∈ ( 0 , 1 ] \delta\in(0,1] δ ∈ ( 0 , 1 ] λ > 1 \lambda> 1 λ > 1 y = { y i } i ∈ N y = \{y_i\}_{i\in\mathbb{N}} y = { y i } i ∈ N ( y i , Π ) (y_i,\Pi) ( y i , Π ) i , π i,\pi i , π y i ( π ) ∈ { 0 , 1 } y_i(\pi) \in\{0,1\} y i ( π ) ∈ { 0 , 1 } y i ( π ) = 1 y_i(\pi) = 1 y i ( π ) = 1 π ≠ ⊥ \pi\ne\bot π = ⊥ x i ( π ) > 0 x_i(\pi) > 0 x i ( π ) > 0
H ( y , Π , ℓ ) ≤ H ( x , Π , ℓ ) , ∀ 0 ≤ ℓ ≤ m , \mathcal{H}(y,\Pi,\ell) \le\mathcal{H}(x,\Pi,\ell), \qquad\forall0 \le\ell\le m, H ( y , Π , ℓ ) ≤ H ( x , Π , ℓ ) , ∀0 ≤ ℓ ≤ m , T ( y , Π , ℓ ) ≤ λ T ( x , Π , ℓ ) , ∀ 0 ≤ ℓ ≤ m , \mathcal{T}(y,\Pi,\ell) \le\lambda\mathcal{T}(x,\Pi,\ell), \qquad\forall0 \le\ell\le m, T ( y , Π , ℓ ) ≤ λ T ( x , Π , ℓ ) , ∀0 ≤ ℓ ≤ m , I ( y , Π ) ≥ I ( x , Π ) − ρ ( q , δ , X ) , \mathcal{I}(y,\Pi) \ge\mathcal{I}(x,\Pi) - \rho(q,\delta,X), I ( y , Π ) ≥ I ( x , Π ) − ρ ( q , δ , X ) , where, for τ ( x , Π ) = ∑ i ∈ N ∑ π ∈ Π x i ( π ) ⋅ τ i ( π ) \tau(x,\Pi) = \sum_{i\in\mathbb{N}}\sum_{\pi\in\Pi} x_i(\pi)\cdot\tau_i(\pi) τ ( x , Π ) = ∑ i ∈ N ∑ π ∈ Π x i ( π ) ⋅ τ i ( π ) X = ∣ { ( i , π ) : 0 < x i ( π ) < 1 } ∣ X = |\{(i,\pi): 0 < x_i(\pi) < 1\}| X = ∣ {( i , π ) : 0 < x i ( π ) < 1 } ∣
ρ ( q , δ , X ) = 8 + 4 log λ max { 1 , 1 δ h max T max X τ ( x , Π ) } + 8 q log max { 1 , X } . \rho(q,\delta,X) = 8 + 4\log\lambda\max\left\{1,\frac{1}{\delta}h_{\max}T_{\max}X\tau(x,\Pi)\right\} + 8q\log\max\{1,X\}. ρ ( q , δ , X ) = 8 + 4 log λ max { 1 , δ 1 h m a x T m a x X τ ( x , Π ) } + 8 q log max { 1 , X } . Here h max : = max i ∈ N h i h_{\max} := \max_{i\in\mathbb{N}} h_i h m a x := max i ∈ N h i T max : = max i ∈ N T i T_{\max} := \max_{i\in\mathbb{N}} T_i T m a x := max i ∈ N T i
We prove Theorem 3.2 in Appendix § 10 (Theorem B.1), similarly to Theorem 2.2.
We will show that combining Lemma 3.2 with Theorem 3.2 guarantees us a set of at least d − O ε , q ( log n ) d-O_{\varepsilon,q}(\log n) d − O ε , q ( log n ) I I I O q ( log n ) O_q(\log n) O q ( log n ) d d d ( d ) (d) ( d ) η \eta η
We resolve this by splitting Theorem 3.1 into two cases, based on the average tail value
T ‾ : = 1 n ∑ i ∈ N T i . \overline{T} := \frac{1}{n}\sum_{i\in\mathbb{N}}T_i. T := n 1 i ∈ N ∑ T i . when T ‾ \overline{T} T ( x i ∗ , Π ) (x_i^*,\Pi) ( x i ∗ , Π ) ⊥ \bot ⊥ ∅ \varnothing ∅ i i i T i > 0 T_i > 0 T i > 0 x ^ \hat{x} x ^ ∅ \varnothing ∅ x ^ \hat{x} x ^ I ( x ^ , Π ) ≥ d + O ε , q ( log n ) \mathcal{I}(\hat{x},\Pi) \ge d + O_{\varepsilon,q}(\log n) I ( x ^ , Π ) ≥ d + O ε , q ( log n ) d d d d d d i ∈ I i \in I i ∈ I π i = ∅ \pi_i = \varnothing π i = ∅ T ‾ \overline{T} T ∅ \varnothing ∅ ∅ \varnothing ∅
the case then T ‾ \overline{T} T M = E + S M = E + S M = E + S ∣ S ∣ = O ε , q ( log n ) |S| = O_{\varepsilon,q}(\log n) ∣ S ∣ = O ε , q ( log n ) S S S E E E d − ∣ S ∣ − O ε , q ( log n ) d - |S| - O_{\varepsilon,q}(\log n) d − ∣ S ∣ − O ε , q ( log n ) E E E S S S d d d E E E S S S S S S
To prove Theorem 3.1 in the case of large T ‾ \overline{T} T
Theorem 3.3 (Large tail). Let γ : = min ( ε , 1 / ( 2 q ) ) \gamma:= \min(\varepsilon,1/(2q)) γ := min ( ε , 1/ ( 2 q )) λ = 2 q 2 q − 1 \lambda= \frac{2q}{2q-1} λ = 2 q − 1 2 q N N N
T ‾ : = 1 n ∑ i ∈ N T i ≥ ρ q , n γ , \overline{T} := \frac{1}{n}\sum_{i\in\mathbb{N}}T_i \ge\frac{\rho_{q,n}}{\gamma}, T := n 1 i ∈ N ∑ T i ≥ γ ρ q , n , where
ρ q , n = 8 + 4 log λ ( q 3 d 2 ( d + 2 ) n ) + 8 q log ( ( d + 2 ) n ) . \rho_{q,n}=8+4\log_{\lambda}\left(q^{3}d^{2}(d+2)n\right)+8q\log\left((d+2)n\right). ρ q , n = 8 + 4 log λ ( q 3 d 2 ( d + 2 ) n ) + 8 q log ( ( d + 2 ) n ) . There exists I ⊆ N I\subseteq N I ⊆ N ∣ I ∣ ≥ d |I|\ge d ∣ I ∣ ≥ d Π I : = { π i } i ∈ I \Pi_I:=\{\pi_i\}_{i\in I} Π I := { π i } i ∈ I i ∈ I i\in I i ∈ I π i \pi_i π i i i i T i = 0 T_i=0 T i = 0 Δ ( π i ) = 0 \Delta(\pi_i)=0 Δ ( π i ) = 0 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m
P ( Π I , ℓ ) ≤ ℓ − ∑ i ∈ I τ i ( π i ) ⋅ ( ℓ − h i ) + . \mathcal{P}(\Pi_I,\ell)\le\ell-\sum_{i\in I}\tau_i(\pi_i)\cdot(\ell-h_i)_+. P ( Π I , ℓ ) ≤ ℓ − i ∈ I ∑ τ i ( π i ) ⋅ ( ℓ − h i ) + . Proof. Construct MMS head profiles π i , j \pi_{i,j} π i , j i ∈ N , j ∈ [ d ] i\in N,j\in[d] i ∈ N , j ∈ [ d ] Π : = { ∅ , ⊥ } ∪ ⋃ i ∈ N { π i , j : j ∈ [ d ] } \Pi:=\{\varnothing,\bot\}\mathbin{\cup}\bigcup_{i\in N}\{\pi_{i,j}:j\in[d]\} Π := { ∅ , ⊥ } ∪ ⋃ i ∈ N { π i , j : j ∈ [ d ]} ( x ∗ , Π ) (x^*,\Pi) ( x ∗ , Π ) ⊥ \bot ⊥ ∅ \varnothing ∅
More specifically, for every agent i ∈ N i\in N i ∈ N ( x ^ i , Π ) (\hat{x}_i,\Pi) ( x ^ i , Π )
x ^ i ( ∅ ) = x i ∗ ( ∅ ) + γ T i n , x ^ i ( ⊥ ) = x i ∗ ( ⊥ ) − γ T i n , ∀ π ∉ { ∅ , ⊥ } : x ^ i ( π ) = x i ∗ ( π ) . \hat{x}_i(\varnothing)=x_i^*(\varnothing)+\frac{\gamma T_i}{n},\qquad
\hat{x}_i(\bot)=x_i^*(\bot)-\frac{\gamma T_i}{n},\qquad
\forall\pi\notin\{\varnothing,\bot\}:\ \hat{x}_i(\pi)=x_i^*(\pi). x ^ i ( ∅ ) = x i ∗ ( ∅ ) + n γ T i , x ^ i ( ⊥ ) = x i ∗ ( ⊥ ) − n γ T i , ∀ π ∈ / { ∅ , ⊥ } : x ^ i ( π ) = x i ∗ ( π ) . These are valid fractional profiles: ∑ π ∈ Π x ^ i = 1 \sum_{\pi\in\Pi}\hat{x}_i=1 ∑ π ∈ Π x ^ i = 1 d ≤ ( 1 − ε ) n d\le(1-\varepsilon)n d ≤ ( 1 − ε ) n T i ≤ v i ( M ) = d T_i\le v_i(M)=d T i ≤ v i ( M ) = d
x ^ i ( ⊥ ) = 1 − d n − γ T i n ≥ ε − γ T i n ≥ γ ( 1 − T i n ) ≥ 0. \hat{x}_i(\bot)=1-\frac{d}{n}-\frac{\gamma T_i}{n}\ge\varepsilon-\frac{\gamma T_i}{n}\ge\gamma\left(1-\frac{T_i}{n}\right)\ge0. x ^ i ( ⊥ ) = 1 − n d − n γ T i ≥ ε − n γ T i ≥ γ ( 1 − n T i ) ≥ 0. Note that only the agents i ∈ N i\in N i ∈ N T i > 0 T_i>0 T i > 0 π = ∅ \pi=\varnothing π = ∅ T i = 0 T_i=0 T i = 0 M M M i i i i i i Δ ( π i , j ) = 0 \Delta(\pi_{i,j})=0 Δ ( π i , j ) = 0 T i = 0 T_i=0 T i = 0 π ∈ Π − ⊥ \pi\in\Pi-\bot π ∈ Π − ⊥ x ^ i ( π ) > 0 \hat{x}_i(\pi)>0 x ^ i ( π ) > 0 Δ ( π ) = 0 \Delta(\pi)=0 Δ ( π ) = 0
It is easy to see that
I ( x ^ , Π ) = ∑ i ∈ N ∑ π ≠ ⊥ x ^ i ( π ) = ∑ i ∈ N ( d n + γ T i n ) = d + γ T ‾ ≥ d + ρ q , n . \mathcal{I}(\hat{x},\Pi)=\sum_{i\in N}\sum_{\pi\ne\bot}\hat{x}_i(\pi)=\sum_{i\in N}\left(\frac{d}{n}+\frac{\gamma T_i}{n}\right)=d+\gamma\overline{T}\ge d+\rho_{q,n}. I ( x ^ , Π ) = i ∈ N ∑ π = ⊥ ∑ x ^ i ( π ) = i ∈ N ∑ ( n d + n γ T i ) = d + γ T ≥ d + ρ q , n . In addition, since P i ( ∅ , ℓ ) = 0 = P i ( ⊥ , ℓ ) \mathcal{P}_i(\varnothing,\ell)=0=\mathcal{P}_i(\bot,\ell) P i ( ∅ , ℓ ) = 0 = P i ( ⊥ , ℓ ) i , ℓ i,\ell i , ℓ H ( x ^ , Π , ℓ ) = H ( x ∗ , Π , ℓ ) \mathcal{H}(\hat{x},\Pi,\ell)=\mathcal{H}(x^*,\Pi,\ell) H ( x ^ , Π , ℓ ) = H ( x ∗ , Π , ℓ ) 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m i ∈ N i\in N i ∈ N T i > 0 T_i>0 T i > 0
τ i ( x ^ i , Π ) = ∑ π ∈ Π x ^ i ( π ) ⋅ τ i ( π ) = 1 n T i ∑ j = 1 d Δ ( π i , j ) + γ T i n ⋅ Δ ( ∅ ) T i ≤ 1 − 1 / q n + γ ( 1 − 1 / q ) n ≤ 1 n ( 1 − 1 2 q − 1 2 q 2 ) ≤ 1 λ n , \begin{aligned}
\tau_i(\hat{x}_i,\Pi)=\sum_{\pi\in\Pi}\hat{x}_i(\pi)\cdot\tau_i(\pi)=\frac{1}{nT_i}\sum_{j=1}^{d}\Delta(\pi_{i,j})+\frac{\gamma T_i}{n}\cdot\frac{\Delta(\varnothing)}{T_i}\le\frac{1-1/q}{n}+\frac{\gamma(1-1/q)}{n} \\
&\le\frac{1}{n}\left(1-\frac{1}{2q}-\frac{1}{2q^2}\right)\le\frac{1}{\lambda n},
\end{aligned} τ i ( x ^ i , Π ) = π ∈ Π ∑ x ^ i ( π ) ⋅ τ i ( π ) = n T i 1 j = 1 ∑ d Δ ( π i , j ) + n γ T i ⋅ T i Δ ( ∅ ) ≤ n 1 − 1/ q + n γ ( 1 − 1/ q ) ≤ n 1 ( 1 − 2 q 1 − 2 q 2 1 ) ≤ λn 1 , using γ ≤ 1 / ( 2 q ) \gamma\le1/(2q) γ ≤ 1/ ( 2 q ) λ = 2 q 2 q − 1 \lambda=\frac{2q}{2q-1} λ = 2 q − 1 2 q τ ( x ^ , Π ) = ∑ i ∈ N τ i ( x ^ i , Π ) ≤ 1 / λ \tau(\hat{x},\Pi)=\sum_{i\in N}\tau_i(\hat{x}_i,\Pi)\le1/\lambda τ ( x ^ , Π ) = ∑ i ∈ N τ i ( x ^ i , Π ) ≤ 1/ λ 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m
T ( x ^ , Π , ℓ ) = ∑ i ∈ N τ i ( x ^ i , Π ) ⋅ ( ℓ − h i ) + ≤ 1 λ n ∑ i ∈ N ( ℓ − h i ) + . \mathcal{T}(\hat{x},\Pi,\ell)=\sum_{i\in N}\tau_i(\hat{x}_i,\Pi)\cdot(\ell-h_i)_+\le\frac{1}{\lambda n}\sum_{i\in N}(\ell-h_i)_+. T ( x ^ , Π , ℓ ) = i ∈ N ∑ τ i ( x ^ i , Π ) ⋅ ( ℓ − h i ) + ≤ λn 1 i ∈ N ∑ ( ℓ − h i ) + . By construction, for every π ∈ Π \pi\in\Pi π ∈ Π ∣ π ∣ ≤ q |\pi|\le q ∣ π ∣ ≤ q π i , j \pi_{i,j} π i , j 1 / q 1/q 1/ q π ∈ Π − ⊥ \pi\in\Pi-\bot π ∈ Π − ⊥ v ( π ) v(\pi) v ( π ) 1 − 1 / q 1-1/q 1 − 1/ q 1 / q 2 1/q^2 1/ q 2 Δ ( π ) = ( 1 − 1 / q − v ( π ) ) > 0 \Delta(\pi)=(1-1/q-v(\pi))>0 Δ ( π ) = ( 1 − 1/ q − v ( π )) > 0 Δ ( π ) ≥ 1 / q 2 \Delta(\pi)\ge1/q^2 Δ ( π ) ≥ 1/ q 2 ( x ^ , Π ) (\hat{x},\Pi) ( x ^ , Π ) λ = 2 q 2 q − 1 \lambda=\frac{2q}{2q-1} λ = 2 q − 1 2 q δ = 1 / q 2 \delta=1/q^2 δ = 1/ q 2 ( y , Π ) (y,\Pi) ( y , Π )
H ( y , Π , ℓ ) ≤ H ( x ^ , Π , ℓ ) , ∀ 0 ≤ ℓ ≤ m , \mathcal{H}(y,\Pi,\ell)\le\mathcal{H}(\hat{x},\Pi,\ell),\qquad\forall0\le\ell\le m, H ( y , Π , ℓ ) ≤ H ( x ^ , Π , ℓ ) , ∀0 ≤ ℓ ≤ m , T ( y , Π , ℓ ) ≤ λ T ( x ^ , Π , ℓ ) , ∀ 0 ≤ ℓ ≤ m , \mathcal{T}(y,\Pi,\ell)\le\lambda\mathcal{T}(\hat{x},\Pi,\ell),\qquad\forall0\le\ell\le m, T ( y , Π , ℓ ) ≤ λ T ( x ^ , Π , ℓ ) , ∀0 ≤ ℓ ≤ m , I ( y , Π ) ≥ I ( x ^ , Π ) − ρ ( q , 1 / q 2 , X ^ ) , \mathcal{I}(y,\Pi)\ge\mathcal{I}(\hat{x},\Pi)-\rho(q,1/q^2,\widehat{X}), I ( y , Π ) ≥ I ( x ^ , Π ) − ρ ( q , 1/ q 2 , X ) , where X ^ = ∣ { ( i , π ) : 0 < x ^ i ( π ) < 1 } ∣ \widehat{X}=|\{(i,\pi):0<\hat{x}_i(\pi)<1\}| X = ∣ {( i , π ) : 0 < x ^ i ( π ) < 1 } ∣ I ⊆ N I \subseteq N I ⊆ N I ( y , Π ) \mathcal{I}(y,\Pi) I ( y , Π ) i ∈ I i \in I i ∈ I y i ( ⊥ ) = 0 y_i(\perp)=0 y i ( ⊥ ) = 0 π i ∈ Π − ⊥ \pi_i \in\Pi-\perp π i ∈ Π − ⊥ y i ( π i ) = 1 y_i(\pi_i)=1 y i ( π i ) = 1 Π I : = { π i } i ∈ I \Pi_I := \{\pi_i\}_{i\in I} Π I := { π i } i ∈ I T i = 0 T_i=0 T i = 0 Δ ( π i ) = 0 \Delta(\pi_i)=0 Δ ( π i ) = 0 T i = 0 T_i=0 T i = 0 x ^ i ( π ) > 0 \hat{x}_i(\pi)>0 x ^ i ( π ) > 0 Δ ( π ) = 0 \Delta(\pi)=0 Δ ( π ) = 0 y i ( π ) > 0 y_i(\pi)>0 y i ( π ) > 0 x i ( π ) > 0 x_i(\pi)>0 x i ( π ) > 0 0 ≤ ℓ ≤ m 0 \le\ell\le m 0 ≤ ℓ ≤ m H ( x ^ , Π , ℓ ) = H ( x ∗ , Π , ℓ ) \mathcal{H}(\hat{x},\Pi,\ell)=\mathcal{H}(x^*,\Pi,\ell) H ( x ^ , Π , ℓ ) = H ( x ∗ , Π , ℓ )
P ( Π I , ℓ ) + ∑ i ∈ I τ i ( π i ) ⋅ ( ℓ − h i ) + = H ( y , Π , ℓ ) + ∑ i ∈ I ∑ π y i ( π ) ⋅ τ i ( π ) ( ℓ − h i ) + = H ( y , Π , ℓ ) + T ( y , Π , ℓ ) ≤ H ( x ^ , Π , ℓ ) + λ T ( x ^ , Π , ℓ ) ≤ ℓ − 1 n ∑ i ∈ N ( ℓ − h i ) + + 1 n ∑ i ∈ N ( ℓ − h i ) + = ℓ . \begin{aligned}
\mathcal{P}(\Pi_I,\ell)+\sum_{i\in I}\tau_i(\pi_i)\cdot(\ell-h_i)_+
=\mathcal{H}(y,\Pi,\ell)+\sum_{i\in I}\sum_{\pi}y_i(\pi)\cdot\tau_i(\pi)(\ell-h_i)_+ \\
&=\mathcal{H}(y,\Pi,\ell)+\mathcal{T}(y,\Pi,\ell)\le\mathcal{H}(\hat{x},\Pi,\ell)+\lambda\mathcal{T}(\hat{x},\Pi,\ell) \\
&\le\ell-\frac{1}{n}\sum_{i\in N}(\ell-h_i)_++\frac{1}{n}\sum_{i\in N}(\ell-h_i)_+=\ell.
\end{aligned} P ( Π I , ℓ ) + i ∈ I ∑ τ i ( π i ) ⋅ ( ℓ − h i ) + = H ( y , Π , ℓ ) + i ∈ I ∑ π ∑ y i ( π ) ⋅ τ i ( π ) ( ℓ − h i ) + = H ( y , Π , ℓ ) + T ( y , Π , ℓ ) ≤ H ( x ^ , Π , ℓ ) + λ T ( x ^ , Π , ℓ ) ≤ ℓ − n 1 i ∈ N ∑ ( ℓ − h i ) + + n 1 i ∈ N ∑ ( ℓ − h i ) + = ℓ . It remains to estimate the number of selected agents ∣ I ∣ |I| ∣ I ∣
∣ I ∣ = I ( y , Π ) ≥ I ( x ^ , Π ) − ρ ( q , 1 / q 2 , X ^ ) ≥ d + ρ q , n − ρ ( q , 1 / q 2 , X ^ ) . |I|=\mathcal{I}(y,\Pi)\ge\mathcal{I}(\hat{x},\Pi)-\rho(q,1/q^2,\hat{X})\ge d+\rho_{q,n}-\rho(q,1/q^2,\hat{X}). ∣ I ∣ = I ( y , Π ) ≥ I ( x ^ , Π ) − ρ ( q , 1/ q 2 , X ^ ) ≥ d + ρ q , n − ρ ( q , 1/ q 2 , X ^ ) . The total number X ^ \hat{X} X ^ x ^ \hat{x} x ^ ( d + 2 ) ⋅ n (d+2)\cdot n ( d + 2 ) ⋅ n d d d π i , j \pi_{i,j} π i , j ∅ \varnothing ∅ ⊥ \perp ⊥ i ∈ N i\in N i ∈ N h i ≤ q d h_i\le qd h i ≤ q d v i ( M ) = d v_i(M)=d v i ( M ) = d 1 / q 1/q 1/ q T i ≤ v i ( M ) ≤ d T_i\le v_i(M)\le d T i ≤ v i ( M ) ≤ d
ρ ( q , 1 / q 2 , X ^ ) = 8 + 4 log λ max { q 2 h max T max X ^ τ ( x ^ , Π ) } + 8 q log max { 1 , X ^ } \rho(q,1/q^2,\hat{X})=8+4\log_{\lambda}\max\left\{q^2h_{\max}T_{\max}\hat{X}\tau(\hat{x},\Pi)\right\}+8q\log\max\{1,\hat{X}\} ρ ( q , 1/ q 2 , X ^ ) = 8 + 4 log λ max { q 2 h m a x T m a x X ^ τ ( x ^ , Π ) } + 8 q log max { 1 , X ^ } ≤ 8 + 4 log λ ( q 3 d 2 ( d + 2 ) n ) + 8 q log ( ( d + 2 ) n ) = ρ q , n . \le8+4\log_{\lambda}\left(q^3d^2(d+2)n\right)+8q\log\left((d+2)n\right)=\rho_{q,n}. ≤ 8 + 4 log λ ( q 3 d 2 ( d + 2 ) n ) + 8 q log ( ( d + 2 ) n ) = ρ q , n . Thus, ∣ I ∣ ≥ d + ρ q , n − ρ ( q , 1 / q 2 , X ^ ) ≥ d |I|\ge d+\rho_{q,n}-\rho(q,1/q^2,\hat{X})\ge d ∣ I ∣ ≥ d + ρ q , n − ρ ( q , 1/ q 2 , X ^ ) ≥ d
Small tail case
When average tail T ‾ \overline{T} T T ‾ ≤ O ε , q ( log n ) \overline{T}\le O_{\varepsilon,q}(\log n) T ≤ O ε , q ( log n ) i ∈ N i\in N i ∈ N ∅ \varnothing ∅ T ‾ = 0 \overline{T}=0 T = 0 ( x ∗ , Π ) (x^*,\Pi) ( x ∗ , Π ) d − O ε , q ( log n ) d-O_{\varepsilon,q}(\log n) d − O ε , q ( log n ) O ε , q ( log n ) O_{\varepsilon,q}(\log n) O ε , q ( log n ) d d d most agents by rounding the fractional MMS profiles. As we will show below, it is actually possible to guarantee something even stronger: Theorem 3.2 can give most agents profiles with deficit 0, while using only a subset of items.
As mentioned earlier, on a high level our approach in the small case is as follows. First, we will split the item set M M M M = E + S M=E+S M = E + S S S S S S S d − O ε , q ( log n ) d-O_{\varepsilon,q}(\log n) d − O ε , q ( log n ) E E E d d d E E E S S S S S S
Partial assignment
First, consider an arbitrary partition M = E + S M=E+S M = E + S π i , j \pi_{i,j} π i , j E E E S S S
Definition 4.1. For every agent i ∈ N i \in N i ∈ N B i = { B i , 1 , … , B i , d } B_i = \{B_{i,1}, \ldots, B_{i,d}\} B i = { B i , 1 , … , B i , d } d d d M M M MMS ( M , v i , d ) \operatorname{MMS}(M,v_i,d) MMS ( M , v i , d ) B i , j ∈ B i B_{i,j} \in B_i B i , j ∈ B i v i ( B i , j ) = 1 v_i(B_{i,j}) = 1 v i ( B i , j ) = 1 M = E + S M = E + S M = E + S j ∈ [ d ] j \in[d] j ∈ [ d ]
π i , j E : = { ⌊ q 2 ⋅ v i ( e ) q 2 ⌋ : e ∈ B i , j ∩ E , v i ( e ) ≥ 1 / q } , \pi_{i,j}^{E} := \left\{ \left\lfloor\frac{q^2 \cdot v_i(e)}{q^2} \right\rfloor: e \in B_{i,j} \cap E, v_i(e) \ge1/q \right\}, π i , j E := { ⌊ q 2 q 2 ⋅ v i ( e ) ⌋ : e ∈ B i , j ∩ E , v i ( e ) ≥ 1/ q } , π i , j S : = { ⌊ q 2 ⋅ v i ( e ) q 2 ⌋ : e ∈ B i , j ∩ S , v i ( e ) ≥ 1 / q } . \pi_{i,j}^{S} := \left\{ \left\lfloor\frac{q^2 \cdot v_i(e)}{q^2} \right\rfloor: e \in B_{i,j} \cap S, v_i(e) \ge1/q \right\}. π i , j S := { ⌊ q 2 q 2 ⋅ v i ( e ) ⌋ : e ∈ B i , j ∩ S , v i ( e ) ≥ 1/ q } . It is clear that for every π i , j \pi_{i,j} π i , j π i , j = π i , j E ⊎ π i , j S \pi_{i,j} = \pi_{i,j}^{E} \uplus\pi_{i,j}^{S} π i , j = π i , j E ⊎ π i , j S ⊎ \uplus ⊎
Let V ⊆ N V \subseteq N V ⊆ N d d d M = E + S M = E + S M = E + S d − ∣ S ∣ − O ε , q ( log n ) d - |S| - O_{\varepsilon,q}(\log n) d − ∣ S ∣ − O ε , q ( log n ) A ⊆ V A \subseteq V A ⊆ V
all profiles have zero deficit (so agents are satisfied already from the head items only);
all profiles are simultaneously satisfiable using distinct items of set E E E
We prove it by observing that, in the small tail case, the number of MMS-profiles with positive deficit is small, and then apply of the rounding Theorem 3.2 to the profiles π i , j \pi_{i,j} π i , j E E E π i , j = π i , j E \pi_{i,j} = \pi_{i,j}^{E} π i , j = π i , j E
Theorem 4.1 (Partial selection). Let M = E + S M = E + S M = E + S V ⊆ N V \subseteq N V ⊆ N ∣ V ∣ ≥ d |V| \ge d ∣ V ∣ ≥ d A ⊆ V A \subseteq V A ⊆ V i ∈ A i \in A i ∈ A j i ∈ [ d ] j_i \in[d] j i ∈ [ d ]
profile π i : = π i , j i \pi_i := \pi_{i,j_i} π i := π i , j i i i i Δ ( π i , j i ) = 0 \Delta(\pi_{i,j_i}) = 0 Δ ( π i , j i ) = 0 π i , j i S = ∅ \pi_{i,j_i}^{S} = \varnothing π i , j i S = ∅ π i , j i E = π i , j i \pi_{i,j_i}^{E} = \pi_{i,j_i} π i , j i E = π i , j i
∑ i ∈ A P i ( π i , ℓ ) ≤ 1 ∣ V ∣ ∑ i ∈ V ∑ j = 1 d P i ( π i , j E , ℓ ) ; \sum_{i\in A} \mathcal{P}_i(\pi_i,\ell) \le\frac{1}{|V|}\sum_{i\in V}\sum_{j=1}^{d} \mathcal{P}_i(\pi_{i,j}^{E},\ell); i ∈ A ∑ P i ( π i , ℓ ) ≤ ∣ V ∣ 1 i ∈ V ∑ j = 1 ∑ d P i ( π i , j E , ℓ ) ; ∣ A ∣ ≥ d − ∣ S ∣ − q ( q − 1 ) ∣ V ∣ ∑ i ∈ V T i − ρ A . |A| \ge d - |S| - \frac{q(q-1)}{|V|}\sum_{i\in V} T_i - \rho_A. ∣ A ∣ ≥ d − ∣ S ∣ − ∣ V ∣ q ( q − 1 ) i ∈ V ∑ T i − ρ A . I.e, when ∣ S ∣ = O ε , q ( log n ) |S| = O_{\varepsilon,q}(\log n) ∣ S ∣ = O ε , q ( log n ) T ‾ = O ε , q ( log n ) \overline{T} = O_{\varepsilon,q}(\log n) T = O ε , q ( log n ) ∣ V ∣ ≥ 3 n / 4 |V| \ge3n/4 ∣ V ∣ ≥ 3 n /4 ∣ A ∣ ≥ d − O ε , q ( log n ) |A| \ge d - O_{\varepsilon,q}(\log n) ∣ A ∣ ≥ d − O ε , q ( log n )
Proof. For every agent i ∈ V i \in V i ∈ V J i J_i J i π i , j \pi_{i,j} π i , j E E E
J i : = { j ∈ [ d ] : Δ ( π i , j ) = 0 and π i , j S = 0 } . J_i := \{j \in[d] : \Delta(\pi_{i,j}) = 0 \text{ and } \pi_{i,j}^{S} = 0\}. J i := { j ∈ [ d ] : Δ ( π i , j ) = 0 and π i , j S = 0 } . First we bound the size of J i J_i J i π i , j S ≠ ∅ \pi_{i,j}^{S} \ne\varnothing π i , j S = ∅ B i , j B_{i,j} B i , j S S S B i , j B_{i,j} B i , j j j j π i , j S ≠ ∅ \pi_{i,j}^{S} \ne\varnothing π i , j S = ∅ d − ∣ S ∣ d-|S| d − ∣ S ∣ v ( π i , j ) v(\pi_{i,j}) v ( π i , j ) 1 − 1 / q 1-1/q 1 − 1/ q 1 / q 2 1/q^2 1/ q 2 Δ ( π i , j ) = ( 1 − 1 / q − v ( π i , k ) ) > 0 \Delta(\pi_{i,j}) = (1-1/q-v(\pi_{i,k})) > 0 Δ ( π i , j ) = ( 1 − 1/ q − v ( π i , k )) > 0 Δ ( π i , j ) ≥ 1 / q 2 \Delta(\pi_{i,j}) \ge1/q^2 Δ ( π i , j ) ≥ 1/ q 2 ∑ j = 1 d Δ ( π i , j ) ≤ ( 1 − 1 / q ) T i \sum_{j=1}^{d}\Delta(\pi_{i,j}) \le(1-1/q)T_i ∑ j = 1 d Δ ( π i , j ) ≤ ( 1 − 1/ q ) T i j ∈ [ d ] j \in[d] j ∈ [ d ] Δ ( π i , j ) > 0 \Delta(\pi_{i,j}) > 0 Δ ( π i , j ) > 0 q 2 ( 1 − 1 / q ) T i = q ( q − 1 ) T i q^2(1-1/q)T_i = q(q-1)T_i q 2 ( 1 − 1/ q ) T i = q ( q − 1 ) T i
∣ J i ∣ ≥ d − ∣ S ∣ − q ( q − 1 ) T i . |J_i| \ge d - |S| - q(q-1)T_i. ∣ J i ∣ ≥ d − ∣ S ∣ − q ( q − 1 ) T i . We now construct the fractional profiles ( x ∗ , Π ) (x^*,\Pi) ( x ∗ , Π ) Π = { ∅ , 1 } + ⋃ i ∈ N { π i , j : j ∈ [ d ] } \Pi= \{\varnothing,\mathbf{1}\} + \bigcup_{i\in N}\{\pi_{i,j}:j\in[d]\} Π = { ∅ , 1 } + ⋃ i ∈ N { π i , j : j ∈ [ d ]} N N N x ∗ x^* x ∗ V V V i i i J i \mathcal{J}_i J i i ∉ V i \notin V i ∈ / V x i ∗ ( ⊥ ) = 1 x_i^*(\perp)=1 x i ∗ ( ⊥ ) = 1 x i ∗ ( π ) = 0 x_i^*(\pi)=0 x i ∗ ( π ) = 0 π ≠ ⊥ \pi\ne\perp π = ⊥ V V V i ∈ V i \in V i ∈ V
x i ∗ ( ⊥ ) = 1 − ∣ J i ∣ ∣ V ∣ , ∀ π ≠ ⊥ : x i ∗ ( π ) = ∣ { j ∈ J i : π i , j = π } ∣ ∣ V ∣ . x_i^*(\perp)=1-\frac{|\mathcal{J}_i|}{|V|}, \qquad\forall\pi\ne\perp:\ x_i^*(\pi)=\frac{|\{j\in\mathcal{J}_i:\pi_{i,j}=\pi\}|}{|V|}. x i ∗ ( ⊥ ) = 1 − ∣ V ∣ ∣ J i ∣ , ∀ π = ⊥: x i ∗ ( π ) = ∣ V ∣ ∣ { j ∈ J i : π i , j = π } ∣ . As ∣ J i ∣ ≤ d ≤ ∣ V ∣ |\mathcal{J}_i|\le d\le|V| ∣ J i ∣ ≤ d ≤ ∣ V ∣
I ( x ∗ , Π ) = ∑ i ∈ N ∑ π ≠ ⊥ x i ∗ ( π ) = ∑ i ∈ V ∣ J i ∣ ∣ V ∣ ≥ d − ∣ S ∣ − q ( q − 1 ) V ∑ i ∈ V T i . \mathcal{I}(x^*,\Pi)=\sum_{i\in N}\sum_{\pi\ne\perp}x_i^*(\pi)=\sum_{i\in V}\frac{|\mathcal{J}_i|}{|V|}\ge d-|S|-\frac{q(q-1)}{V}\sum_{i\in V}T_i. I ( x ∗ , Π ) = i ∈ N ∑ π = ⊥ ∑ x i ∗ ( π ) = i ∈ V ∑ ∣ V ∣ ∣ J i ∣ ≥ d − ∣ S ∣ − V q ( q − 1 ) i ∈ V ∑ T i . Next, let i ∈ V i\in V i ∈ V 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m α ∈ π i , j \alpha\in\pi_{i,j} α ∈ π i , j j ∈ J i j\in\mathcal{J}_i j ∈ J i e ∈ B i , j e\in B_{i,j} e ∈ B i , j α \alpha α E E E α \alpha α P i ( π i , j , ℓ ) = ∣ { α ∈ π i , j : m i ( α ) ≤ ℓ } ∣ \mathcal{P}_i(\pi_{i,j},\ell)=|\{\alpha\in\pi_{i,j}:m_i(\alpha)\le\ell\}| P i ( π i , j , ℓ ) = ∣ { α ∈ π i , j : m i ( α ) ≤ ℓ } ∣ ℓ \ell ℓ e e e α \alpha α { 1 , … , ℓ } \{1,\ldots,\ell\} { 1 , … , ℓ } e e e B i , j B_{i,j} B i , j
∑ j ∈ J i P i ( π i , j , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣ . \sum_{j\in\mathcal{J}_i}\mathcal{P}_i(\pi_{i,j},\ell)\le|E\cap[\ell]|. j ∈ J i ∑ P i ( π i , j , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣. Averaging over V V V 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m ℓ \ell ℓ x ∗ x^* x ∗
H ( x ∗ , Π , ℓ ) = 1 ∣ V ∣ ∑ i ∈ V ∑ j ∈ J i P i ( π i , j , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣ . \mathcal{H}(x^*,\Pi,\ell)=\frac{1}{|V|}\sum_{i\in V}\sum_{j\in\mathcal{J}_i}\mathcal{P}_i(\pi_{i,j},\ell)\le|E\cap[\ell]|. H ( x ∗ , Π , ℓ ) = ∣ V ∣ 1 i ∈ V ∑ j ∈ J i ∑ P i ( π i , j , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣. As all deficits are zero, ( x ∗ , Π ) (x^*,\Pi) ( x ∗ , Π ) τ ( x , Π ) = 0 \tau(x,\Pi)=0 τ ( x , Π ) = 0 T ( x ∗ , Π , ℓ ) = 0 \mathcal{T}(x^*,\Pi,\ell)=0 T ( x ∗ , Π , ℓ ) = 0 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m ( x ∗ , Π ) (x^*,\Pi) ( x ∗ , Π ) X ∗ X^* X ∗ x ∗ x^* x ∗ ( d + 1 ) ⋅ ∣ V ∣ ≤ ( d + 1 ) n ≤ n 2 (d+1)\cdot|V|\le(d+1)n\le n^2 ( d + 1 ) ⋅ ∣ V ∣ ≤ ( d + 1 ) n ≤ n 2 ( y , Π ) (y,\Pi) ( y , Π ) ρ ( q , 1 / q 2 , X ∗ ) ≤ 8 + 16 q log n = : ρ A \rho(q,1/q^2,X^*)\le8+16q\log n=:\rho_A ρ ( q , 1/ q 2 , X ∗ ) ≤ 8 + 16 q log n =: ρ A
∀ 0 ≤ ℓ ≤ m : H ( y , Π , ℓ ) ≤ H ( x ∗ , Π , ℓ ) and I ( y , Π ) ≥ I ( x , Π ) − ρ A . \forall0\le\ell\le m:\ \mathcal{H}(y,\Pi,\ell)\le\mathcal{H}(x^*,\Pi,\ell)\qquad\text{and}\qquad\mathcal{I}(y,\Pi)\ge\mathcal{I}(x,\Pi)-\rho_A. ∀0 ≤ ℓ ≤ m : H ( y , Π , ℓ ) ≤ H ( x ∗ , Π , ℓ ) and I ( y , Π ) ≥ I ( x , Π ) − ρ A . Let A = { i ∈ N : y i ( ⊥ ) = 0 } A=\{i\in N:y_i(\perp)=0\} A = { i ∈ N : y i ( ⊥ ) = 0 } I ( y , Π ) \mathcal{I}(y,\Pi) I ( y , Π ) y y y x ∗ x^* x ∗ A ⊆ V A\subseteq V A ⊆ V π i ∈ Π \pi_i\in\Pi π i ∈ Π y i ( π i ) = 1 y_i(\pi_i)=1 y i ( π i ) = 1 π i = π i , j i \pi_i=\pi_{i,j_i} π i = π i , j i j i ∈ J i j_i\in\mathcal{J}_i j i ∈ J i Π A : = { π i , j i : i ∈ A } \Pi_A:=\{\pi_{i,j_i}:i\in A\} Π A := { π i , j i : i ∈ A } 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m
P ( Π A , ℓ ) = H ( y , Π , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣ , \mathcal{P}(\Pi_A,\ell)=\mathcal{H}(y,\Pi,\ell)\le|E\cap[\ell]|, P ( Π A , ℓ ) = H ( y , Π , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣ , so by Lemma 2.2 we can satisfy Π A \Pi_A Π A E E E 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m
∑ i ∈ A P i ( π i , ℓ ) ≤ 1 ∣ V ∣ ∑ i ∈ V ∑ j = 1 d P i ( π i , j E , ℓ ) . \sum_{i\in A}\mathcal{P}_i(\pi_i,\ell)\le\frac{1}{|V|}\sum_{i\in V}\sum_{j=1}^{d}\mathcal{P}_i(\pi_{i,j}^{E},\ell). i ∈ A ∑ P i ( π i , ℓ ) ≤ ∣ V ∣ 1 i ∈ V ∑ j = 1 ∑ d P i ( π i , j E , ℓ ) . Consider some partition M = E + S M=E+S M = E + S V ⊆ N V\subseteq N V ⊆ N A ⊆ V A\subseteq V A ⊆ V ∣ A ∣ ≥ d |A|\ge d ∣ A ∣ ≥ d E E E V V V ∣ A ∣ < d |A|<d ∣ A ∣ < d ∣ S ∣ |S| ∣ S ∣ T ‾ \overline{T} T O ε , q ( log n ) O_{\varepsilon,q}(\log n) O ε , q ( log n ) d − ∣ A ∣ ≤ O ε , q ( log n ) d-|A|\le O_{\varepsilon,q}(\log n) d − ∣ A ∣ ≤ O ε , q ( log n ) O ε , q ( log n ) O_{\varepsilon,q}(\log n) O ε , q ( log n ) d d d
Before applying Theorem 4.1, we will fix some set U ⊆ N − V U\subseteq N-V U ⊆ N − V A ⊆ V A\subseteq V A ⊆ V A 0 ⊆ A A_0\subseteq A A 0 ⊆ A { π i } i ∈ A 0 \{\pi_i\}_{i\in A_0} { π i } i ∈ A 0 U 0 ⊆ U U_0\subseteq U U 0 ⊆ U { π u } u ∈ U 0 \{\pi_u\}_{u\in U_0} { π u } u ∈ U 0 A 0 , U 0 A_0,U_0 A 0 , U 0 ∣ A 0 ∣ + ∣ U 0 ∣ = d |A_0|+|U_0|=d ∣ A 0 ∣ + ∣ U 0 ∣ = d 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m
∑ i ∈ A 0 P i ( π i , ℓ ) + ∑ u ∈ U 0 P u ( π u , ℓ ) ≤ ℓ . \sum_{i\in A_0}\mathcal{P}_i(\pi_i,\ell)+\sum_{u\in U_0}\mathcal{P}_u(\pi_u,\ell)\le\ell. i ∈ A 0 ∑ P i ( π i , ℓ ) + u ∈ U 0 ∑ P u ( π u , ℓ ) ≤ ℓ . For now we will not worry about additional slackness for tail completion, this will be handled later. The main question is, even without tails, how to select agents U 0 U_0 U 0 A 0 , U 0 A_0,U_0 A 0 , U 0
The key point is that agents A 0 A_0 A 0 E E E E E E U 0 U_0 U 0 A 0 A_0 A 0 U 0 U_0 U 0 S = M − E S=M-E S = M − E U 0 U_0 U 0 S S S S S S M = E + S M=E+S M = E + S N = U + V + W N=U+V+W N = U + V + W U 0 ⊆ U U_0\subseteq U U 0 ⊆ U
Saved items
Suppose that we selected agents A 0 ⊆ A A_0\subseteq A A 0 ⊆ A U 0 ⊆ U U_0\subseteq U U 0 ⊆ U { π u } u ∈ U 0 \{\pi_u\}_{u\in U_0} { π u } u ∈ U 0 ∣ A 0 ∣ + ∣ U 0 ∣ = d |A_0|+|U_0|=d ∣ A 0 ∣ + ∣ U 0 ∣ = d A 0 , U 0 A_0,U_0 A 0 , U 0 0 ≤ ℓ ≤ m 0\leq\ell\leq m 0 ≤ ℓ ≤ m
∑ i ∈ A 0 P i ( π i , ℓ ) + ∑ u ∈ U 0 P u ( π u , ℓ ) ≤ ℓ . \sum_{i\in A_0}\mathcal{P}_i(\pi_i,\ell)+\sum_{u\in U_0}\mathcal{P}_u(\pi_u,\ell)\leq\ell. i ∈ A 0 ∑ P i ( π i , ℓ ) + u ∈ U 0 ∑ P u ( π u , ℓ ) ≤ ℓ . Jumping ahead, for every agent u ∈ U 0 u\in U_0 u ∈ U 0 π u ∗ , j \pi_{u_*,j} π u ∗ , j u ∗ ∈ U u_*\in U u ∗ ∈ U u ∗ = u u_*=u u ∗ = u j ∈ [ d ] j\in[d] j ∈ [ d ] π u : = π u E ⊎ π u S \pi_u:=\pi_u^E\mathbin{\uplus}\pi_u^S π u := π u E ⊎ π u S u ∈ U 0 u\in U_0 u ∈ U 0
Observe that, for the demands of A 0 A_0 A 0 U 0 U_0 U 0 ℓ \ell ℓ E E E E ∩ [ ℓ ] E\cap[\ell] E ∩ [ ℓ ] S S S S ∩ [ ℓ ] S\cap[\ell] S ∩ [ ℓ ] i ∈ A 0 i\in A_0 i ∈ A 0 π i = π i , j i \pi_i=\pi_{i,j_i} π i = π i , j i π i , j i S = ∅ \pi_{i,j_i}^S=\varnothing π i , j i S = ∅ π i = π i E \pi_i=\pi_i^E π i = π i E E ∩ [ ℓ ] E\cap[\ell] E ∩ [ ℓ ] E E E A 0 , U 0 A_0,U_0 A 0 , U 0 S S S U 0 U_0 U 0 0 ≤ ℓ ≤ m 0\leq\ell\leq m 0 ≤ ℓ ≤ m
∑ i ∈ A 0 P i ( π i E , ℓ ) + ∑ u ∈ U 0 P u ( π u E , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣ ; \sum_{i\in A_0}\mathcal{P}_i(\pi_i^E,\ell)+\sum_{u\in U_0}\mathcal{P}_u(\pi_u^E,\ell)\leq|E\cap[\ell]|; i ∈ A 0 ∑ P i ( π i E , ℓ ) + u ∈ U 0 ∑ P u ( π u E , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣ ; ∑ u ∈ U 0 P u ( π u S , ℓ ) ≤ ∣ S ∩ [ ℓ ] ∣ . \sum_{u\in U_0}\mathcal{P}_u(\pi_u^S,\ell)\leq|S\cap[\ell]|. u ∈ U 0 ∑ P u ( π u S , ℓ ) ≤ ∣ S ∩ [ ℓ ] ∣. Since M = E + S M=E+S M = E + S P u ( π u , ℓ ) = P u ( π u E , ℓ ) + P u ( π u S , ℓ ) \mathcal{P}_u(\pi_u,\ell)=\mathcal{P}_u(\pi_u^E,\ell)+\mathcal{P}_u(\pi_u^S,\ell) P u ( π u , ℓ ) = P u ( π u E , ℓ ) + P u ( π u S , ℓ )
∑ i ∈ A 0 P i ( π i , ℓ ) + ∑ u ∈ U 0 P u ( π u , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣ + ∣ S ∩ [ ℓ ] ∣ = ℓ . \sum_{i\in A_0}\mathcal{P}_i(\pi_i,\ell)+\sum_{u\in U_0}\mathcal{P}_u(\pi_u,\ell)\leq|E\cap[\ell]|+|S\cap[\ell]|=\ell. i ∈ A 0 ∑ P i ( π i , ℓ ) + u ∈ U 0 ∑ P u ( π u , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣ + ∣ S ∩ [ ℓ ] ∣ = ℓ . We will first focus on condition (E). Theorem 4.1 guarantees us that for every 0 ≤ ℓ ≤ m 0\leq\ell\leq m 0 ≤ ℓ ≤ m
∑ i ∈ A 0 P i ( π i E , ℓ ) ≤ 1 ∣ V ∣ ∑ i ∈ V ∑ j = 1 d P i ( π i , j E , ℓ ) . \sum_{i\in A_0}\mathcal{P}_i(\pi_i^E,\ell)\leq\frac{1}{|V|}\sum_{i\in V}\sum_{j=1}^{d}\mathcal{P}_i(\pi_{i,j}^E,\ell). i ∈ A 0 ∑ P i ( π i E , ℓ ) ≤ ∣ V ∣ 1 i ∈ V ∑ j = 1 ∑ d P i ( π i , j E , ℓ ) . We would like to pick sets E ⊆ M E\subseteq M E ⊆ M V ⊆ N V\subseteq N V ⊆ N ∣ E ∩ [ ℓ ] ∣ |E\cap[\ell]| ∣ E ∩ [ ℓ ] ∣ ∑ u ∈ U 0 P u ( π u E , ℓ ) \sum_{u\in U_0}\mathcal{P}_u(\pi_u^E,\ell) ∑ u ∈ U 0 P u ( π u E , ℓ ) U 0 U_0 U 0 0 ≤ ℓ ≤ m 0\leq\ell\leq m 0 ≤ ℓ ≤ m
∑ u ∈ U 0 P u ( π u E , ℓ ) ≤ ∑ u ∈ U 0 ∣ π u E ∣ ≤ q ∣ U 0 ∣ , \sum_{u\in U_0}\mathcal{P}_u(\pi_u^E,\ell)\leq\sum_{u\in U_0}|\pi_u^E|\leq q|U_0|, u ∈ U 0 ∑ P u ( π u E , ℓ ) ≤ u ∈ U 0 ∑ ∣ π u E ∣ ≤ q ∣ U 0 ∣ , since MMS profiles have at most q q q E ⊆ M E\subseteq M E ⊆ M V ⊆ N V\subseteq N V ⊆ N U 0 ⊆ U U_0\subseteq U U 0 ⊆ U 0 ≤ ℓ ≤ m 0\leq\ell\leq m 0 ≤ ℓ ≤ m ℓ \ell ℓ
1 ∣ V ∣ ∑ i ∈ V ∑ j = 1 d P i ( π i , j E , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣ − q ∣ U 0 ∣ . (1) \frac{1}{|V|}\sum_{i\in V}\sum_{j=1}^{d}\mathcal{P}_i(\pi_{i,j}^E,\ell)\leq|E\cap[\ell]|-q|U_0|. \tag*{(1)} ∣ V ∣ 1 i ∈ V ∑ j = 1 ∑ d P i ( π i , j E , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣ − q ∣ U 0 ∣. (1) This condition turns out to be too strong. While the demands of P i , j E \mathcal{P}^{E}_{i,j} P i , j E E E E need not have their breakpoints in E E E ℓ \ell ℓ ℓ ∈ S \ell\in S ℓ ∈ S S S S S S S L S ⊂ S L_S \subset S L S ⊂ S protected prefixes , of size at most 2 q 2 2q^2 2 q 2 ℓ s ∈ L S \ell_s \in L_S ℓ s ∈ L S
∑ i ∈ A 0 P i E ( π , ℓ s ) + ∑ u ∈ U 0 P u E ( π , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣ . (2) \sum_{i\in A_0}\mathcal{P}_{i}^{E}(\pi,\ell_s)+\sum_{u\in U_0}\mathcal{P}_{u}^{E}(\pi,\ell_s)\le|E\cap[\ell_s]|. \tag*{(2)} i ∈ A 0 ∑ P i E ( π , ℓ s ) + u ∈ U 0 ∑ P u E ( π , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣. (2) The sets S S S L S L_S L S ℓ s ∈ L S \ell_s \in L_S ℓ s ∈ L S η ≃ q 2 q 2 ∣ S ∣ \eta\simeq q^{2q^2}|S| η ≃ q 2 q 2 ∣ S ∣ ℓ ∈ E \ell\in E ℓ ∈ E breakpoint of some agent i ∈ V i \in V i ∈ V
1 ∣ V ∣ ∑ i ∈ V ∑ j = 1 d P i ( π i , j E , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣ − η 3 . \frac{1}{|V|}\sum_{i\in V}\sum_{j=1}^{d}\mathcal{P}_{i}(\pi^{E}_{i,j},\ell)\le|E\cap[\ell]|-\frac{\eta}{3}. ∣ V ∣ 1 i ∈ V ∑ j = 1 ∑ d P i ( π i , j E , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣ − 3 η . We will show that, with this choice of S S S L S L_S L S 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m U 0 ⊆ U U_0\subseteq U U 0 ⊆ U ∣ U 0 ∣ ≤ η / ( 3 q ) |U_0|\le\eta/(3q) ∣ U 0 ∣ ≤ η / ( 3 q ) ℓ s ∈ L S \ell_s\in L_S ℓ s ∈ L S ∣ U 0 ∣ = d − ∣ A 0 ∣ ≥ d − ∣ A ∣ |U_0|=d-|A_0|\ge d-|A| ∣ U 0 ∣ = d − ∣ A 0 ∣ ≥ d − ∣ A ∣ ∣ S ∣ |S| ∣ S ∣ T ‾ \overline{T} T O ε , q ( log n ) O_{\varepsilon,q}(\log n) O ε , q ( log n ) d − ∣ A ∣ ≤ c ε , q log n d-|A|\le c_{\varepsilon,q}\log n d − ∣ A ∣ ≤ c ε , q log n c ε , q c_{\varepsilon,q} c ε , q U 0 U_0 U 0 c ε , q log n ≤ ∣ U 0 ∣ ≤ η / 3 q = O ε , q ( ∣ S ∣ ) c_{\varepsilon,q}\log n\le|U_0|\le\eta/3q=O_{\varepsilon,q}(|S|) c ε , q log n ≤ ∣ U 0 ∣ ≤ η /3 q = O ε , q ( ∣ S ∣ ) n n n U 0 U_0 U 0
Construction of item/agent partitions
Our set S ⊆ M S\subseteq M S ⊆ M
S = { a 1 , a 1 + 1 , … , b 1 } ∪ … ∪ { a t , a t + 1 , … , b t } . S=\{a_1,a_1+1,\ldots,b_1\}\cup\ldots\cup\{a_t,a_t+1,\ldots,b_t\}. S = { a 1 , a 1 + 1 , … , b 1 } ∪ … ∪ { a t , a t + 1 , … , b t } . where each [ a s , b s ] [a_s,b_s] [ a s , b s ] s ∈ [ t ] s\in[t] s ∈ [ t ] S S S t ≤ 2 q 2 t\le2q^2 t ≤ 2 q 2 S S S O ε , q ( log n ) O_{\varepsilon,q}(\log n) O ε , q ( log n ) a 1 = 1 a_1=1 a 1 = 1 S S S { 1 , … , b 1 } \{1,\ldots,b_1\} { 1 , … , b 1 } L S L_S L S b s b_s b s s ∈ [ t ] s\in[t] s ∈ [ t ]
Put E = M − S E=M-S E = M − S S S S breakpoint-free region F ⊆ E F\subseteq E F ⊆ E S S S [ a s , b s ] ⊆ S [a_s,b_s]\subseteq S [ a s , b s ] ⊆ S s ∈ [ t ] s\in[t] s ∈ [ t ] { b s + 1 , … , b s + η } \{b_s+1,\ldots,b_s+\eta\} { b s + 1 , … , b s + η } η ≥ q 2 q 2 ∣ S ∣ \eta\ge q^{2q^2}|S| η ≥ q 2 q 2 ∣ S ∣
F = E ∩ ( { b 1 + 1 , … , b 1 + η } ∪ … ∪ { b t + 1 , … , b t + η } ) . F=E\cap(\{b_1+1,\ldots,b_1+\eta\}\cup\ldots\cup\{b_t+1,\ldots,b_t+\eta\}). F = E ∩ ({ b 1 + 1 , … , b 1 + η } ∪ … ∪ { b t + 1 , … , b t + η }) . The set S S S η \eta η
the set of agents W ⊆ N W\subseteq N W ⊆ N F F F ∣ W ∣ ≤ ε n / 8 |W|\le\varepsilon n/8 ∣ W ∣ ≤ ε n /8
if ℓ ∈ E − F \ell\in E-F ℓ ∈ E − F N − W N-W N − W ( ℓ − η , ℓ ] = { ℓ − η + 1 , … , ℓ } (\ell-\eta,\ell]=\{\ell-\eta+1,\ldots,\ell\} ( ℓ − η , ℓ ] = { ℓ − η + 1 , … , ℓ } E E E n / 2 n/2 n /2 ( ℓ − η , ℓ ] (\ell-\eta,\ell] ( ℓ − η , ℓ ]
In a sense, set of saved items S S S F ⊆ E F\subseteq E F ⊆ E E E E E E E N − W N-W N − W E E E V ⊆ N − W V\subseteq N-W V ⊆ N − W η / 3 \eta/3 η /3
We will also have partition agents N N N N = U + V + W N=U+V+W N = U + V + W
W W W F F F ∣ W ∣ ≤ ε n / 8 |W|\le\varepsilon n/8 ∣ W ∣ ≤ ε n /8 W W W
U U U ε n / 8 \varepsilon n/8 ε n /8 N − W N-W N − W largest heads h i h_i h i U U U
V = N − U − W V = N - U - W V = N − U − W ∣ V ∣ ≥ ( 1 − ε / 4 ) n |V| \ge(1 - \varepsilon/4)n ∣ V ∣ ≥ ( 1 − ε /4 ) n E = M − S E = M - S E = M − S A ⊆ V A \subseteq V A ⊆ V
We will now establishes our choices of set W ⊂ N W \subset N W ⊂ N S ⊂ M S \subset M S ⊂ M Q Q Q
Q : = { q + k q 2 : k ∈ { 0 , 1 , 2 , … , q 2 − q } } . Q := \left\{ \frac{q+k}{q^2} : k \in\{0,1,2,\ldots,q^2-q\} \right\}. Q := { q 2 q + k : k ∈ { 0 , 1 , 2 , … , q 2 − q } } . Lemma 4.1 (Saved items). Let ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 ) r 1 , C 1 ≥ 2 r_1, C_1 \ge2 r 1 , C 1 ≥ 2 r 1 ≤ n r_1 \le n r 1 ≤ n W ⊂ N W \subset N W ⊂ N ∣ W ∣ ≤ ε n / 8 |W| \le\varepsilon n/8 ∣ W ∣ ≤ ε n /8 S ⊂ M S \subset M S ⊂ M r 1 ≤ ∣ S ∣ ≤ 2 q 2 C 1 8 q 2 / ε r 1 r_1 \le|S| \le2q^2 C_1^{8q^2/\varepsilon}r_1 r 1 ≤ ∣ S ∣ ≤ 2 q 2 C 1 8 q 2 / ε r 1
S = { a 1 , a 1 + 1 , … , b 1 } ∪ ⋯ ∪ { a t , a t + 1 , … , b t } S = \{a_1,a_1+1,\ldots,b_1\} \cup\cdots\cup\{a_t,a_t+1,\ldots,b_t\} S = { a 1 , a 1 + 1 , … , b 1 } ∪ ⋯ ∪ { a t , a t + 1 , … , b t } for t ≤ 2 q 2 t \le2q^2 t ≤ 2 q 2 1 ∈ S 1 \in S 1 ∈ S [ a s , b s ] ⊂ S [a_s,b_s] \subset S [ a s , b s ] ⊂ S s ∈ [ t ] s \in[t] s ∈ [ t ] E = M − S E = M - S E = M − S η \eta η
η ≥ C 1 − 1 2 q 2 ∣ S ∣ , \eta\ge\frac{C_1-1}{2q^2}|S|, η ≥ 2 q 2 C 1 − 1 ∣ S ∣ , such that for every i ∈ N − W i \in N - W i ∈ N − W α ∈ Q \alpha\in Q α ∈ Q ℓ : = m i ( α ) \ell:= m_i(\alpha) ℓ := m i ( α ) E E E
ℓ ≥ η \ell\ge\eta ℓ ≥ η η \eta η E E E { ℓ − η + 1 , … , ℓ } ⊆ E \{\ell-\eta+1,\ldots,\ell\} \subseteq E { ℓ − η + 1 , … , ℓ } ⊆ E
the interval ( ℓ − η , ℓ ] (\ell-\eta,\ell] ( ℓ − η , ℓ ] n / 2 n/2 n /2
∣ { ( i , α ) : i ∈ N , α ∈ Q and ℓ − η < m i ( α ) ≤ ℓ } ∣ ≤ n / 2. |\{(i,\alpha): i \in N,\ \alpha\in Q\ \text{and}\ \ell-\eta< m_i(\alpha) \le\ell\}| \le n/2. ∣ {( i , α ) : i ∈ N , α ∈ Q and ℓ − η < m i ( α ) ≤ ℓ } ∣ ≤ n /2. Proof. For every agent i ∈ N i \in N i ∈ N i i i m i ( α ) > 0 m_i(\alpha)>0 m i ( α ) > 0 α ∈ Q \alpha\in Q α ∈ Q M ( Q ) \mathcal{M}(Q) M ( Q )
M ( Q ) : = ⨄ i ∈ N α ∈ Q m i ( α ) > 0 { ( i , m i ( α ) ) } . \mathcal{M}(Q) := \biguplus_{\substack{i\in N\\ \alpha\in Q\\m_i(\alpha)>0}}\{(i,m_i(\alpha))\}. M ( Q ) := i ∈ N α ∈ Q m i ( α ) > 0 ⨄ {( i , m i ( α ))} . Since ∣ Q ∣ ≤ q 2 |Q| \le q^2 ∣ Q ∣ ≤ q 2 ∣ M ( Q ) ∣ ≤ q 2 n |\mathcal{M}(Q)| \le q^2n ∣ M ( Q ) ∣ ≤ q 2 n M ( Q ) \mathcal{M}(Q) M ( Q ) m i ( α ) m_i(\alpha) m i ( α ) ( ⌊ n / 2 ⌋ + 1 ) (\lfloor n/2\rfloor+1) (⌊ n /2 ⌋ + 1 ) marked breakpoints
M 0 = { ( i k ( ⌊ n / 2 ⌋ + 1 ) , m k ( ⌊ n / 2 ⌋ + 1 ) ) : k = 1 , 2 , … } . \mathcal{M}_0 = \{(i_{k(\lfloor n/2\rfloor+1)},m_{k(\lfloor n/2\rfloor+1)}): k=1,2,\ldots\}. M 0 = {( i k (⌊ n /2 ⌋ + 1 ) , m k (⌊ n /2 ⌋ + 1 ) ) : k = 1 , 2 , … } . Clearly, ∣ M 0 ∣ ≤ 2 q 2 − 1 |\mathcal{M}_0| \le2q^2-1 ∣ M 0 ∣ ≤ 2 q 2 − 1 M ( Q ) \mathcal{M}(Q) M ( Q ) n / 2 n/2 n /2 S 0 S_0 S 0 { 1 } \{1\} { 1 } M M M M 0 \mathcal{M}_0 M 0
S 0 = { 1 } ∪ { s ∈ { 1 , … , m } : ( i , s ) ∈ M 0 for some i ∈ N } . S_0 = \{1\} \cup\{s \in\{1,\ldots,m\} : (i,s) \in\mathcal{M}_0\text{ for some }i\in N\}. S 0 = { 1 } ∪ { s ∈ { 1 , … , m } : ( i , s ) ∈ M 0 for some i ∈ N } . As ∣ M 0 ∣ ≤ 2 q 2 − 1 |\mathcal{M}_0| \le2q^2-1 ∣ M 0 ∣ ≤ 2 q 2 − 1 ∣ S 0 ∣ ≤ 2 q 2 |S_0| \le2q^2 ∣ S 0 ∣ ≤ 2 q 2
We will now construct a sequence of saved regions . For integer radius r ≥ 1 r \ge1 r ≥ 1 S ( r ) S(r) S ( r ) M M M r r r S 0 S_0 S 0
S ( r ) : = ⋃ s ∈ S 0 { s , s + 1 , … , s + r − 1 } ∩ M . S(r) := \bigcup_{s\in S_0}\{s,s+1,\ldots,s+r-1\}\cap M. S ( r ) := s ∈ S 0 ⋃ { s , s + 1 , … , s + r − 1 } ∩ M . We will also have a sequence of protected regions F ( r ) : = S ( C 1 ⋅ r ) − S ( r ) F(r) := S(C_1\cdot r)-S(r) F ( r ) := S ( C 1 ⋅ r ) − S ( r )
M F ( r ) : = { ( i k , m i k ) ∈ M ( Q ) : m i k ∈ F ( r ) } \mathcal{M}_F(r) := \{(i_k,m_{i_k}) \in\mathcal{M}(Q):m_{i_k}\in F(r)\} M F ( r ) := {( i k , m i k ) ∈ M ( Q ) : m i k ∈ F ( r )} be the multiset of all breakpoints in M ( Q ) \mathcal{M}(Q) M ( Q ) F ( r ) F(r) F ( r )
We start with S ( r 1 ) S(r_1) S ( r 1 ) r 1 r_1 r 1 k ≥ 1 k \ge1 k ≥ 1 C 1 C_1 C 1 r k + 1 = C 1 ⋅ r k r_{k+1}=C_1\cdot r_k r k + 1 = C 1 ⋅ r k S ( r k + 1 ) = S ( r k ) + F ( r k ) S(r_{k+1})=S(r_k)+F(r_k) S ( r k + 1 ) = S ( r k ) + F ( r k ) r k → r k + 1 r_k \to r_{k+1} r k → r k + 1 k ∗ k^* k ∗ ⌊ ε n / 8 ⌋ \lfloor\varepsilon n/8\rfloor ⌊ ε n /8 ⌋ k ∗ k^* k ∗ F ( r k ∗ ) F(r_{k^*}) F ( r k ∗ )
∣ M F ( r k ∗ ) ∣ ≤ ⌊ ε n / 8 ⌋ . |\mathcal{M}_{F}(r_{k^*})|\le\lfloor\varepsilon n/8\rfloor. ∣ M F ( r k ∗ ) ∣ ≤ ⌊ ε n /8 ⌋ . Since ∣ M ( Q ) ∣ ≤ q 2 n |\mathcal{M}(Q)|\le q^2n ∣ M ( Q ) ∣ ≤ q 2 n ⌊ 8 q 2 / ε ⌋ \lfloor8q^2/\varepsilon\rfloor ⌊ 8 q 2 / ε ⌋ r ∗ : = r k ∗ r_*:=r_{k^*} r ∗ := r k ∗
r ∗ ≤ C 1 8 q 2 / ε r 1 . r_*\le C_1^{8q^2/\varepsilon}r_1. r ∗ ≤ C 1 8 q 2 / ε r 1 . For this r ∗ r_* r ∗ S S S S = S ( r ∗ ) S=S(r_*) S = S ( r ∗ ) F F F F = F ( r ∗ ) F=F(r_*) F = F ( r ∗ ) W W W
W = { i ∈ N : ∃ α ∈ Q such that m i ( α ) ∈ F } . W=\{i\in N:\exists\alpha\in Q\text{ such that }m_i(\alpha)\in F\}. W = { i ∈ N : ∃ α ∈ Q such that m i ( α ) ∈ F } . Clearly, ∣ W ∣ ≤ ∣ M F ( r ∗ ) ∣ ≤ ε n / 8 |W|\le|\mathcal{M}_{F}(r_*)|\le\varepsilon n/8 ∣ W ∣ ≤ ∣ M F ( r ∗ ) ∣ ≤ ε n /8
S = ⋃ s ∈ S 0 { s , s + 1 , … , s + r ∗ − 1 } ∩ M , S=\bigcup_{s\in S_0}\{s,s+1,\ldots,s+r_*-1\}\cap M, S = s ∈ S 0 ⋃ { s , s + 1 , … , s + r ∗ − 1 } ∩ M , so S S S ∣ S 0 ∣ ≤ 2 q 2 |S_0|\le2q^2 ∣ S 0 ∣ ≤ 2 q 2 r ∗ r_* r ∗ S S S S ( r 1 ) S(r_1) S ( r 1 ) r 1 ≤ ∣ S ∣ ≤ 2 q 2 r ∗ ≤ 2 q 2 C 1 8 q 2 / ε r 1 r_1\le|S|\le2q^2r_*\le2q^2C_1^{8q^2/\varepsilon}r_1 r 1 ≤ ∣ S ∣ ≤ 2 q 2 r ∗ ≤ 2 q 2 C 1 8 q 2 / ε r 1 η : = ( C 1 − 1 ) ⋅ r ∗ \eta:=(C_1-1)\cdot r_* η := ( C 1 − 1 ) ⋅ r ∗
η ≥ C 1 − 1 2 q 2 ∣ S ∣ . \eta\ge\frac{C_1-1}{2q^2}|S|. η ≥ 2 q 2 C 1 − 1 ∣ S ∣. We will now prove that integer η = ( C 1 − 1 ) ⋅ r ∗ \eta=(C_1-1)\cdot r_* η = ( C 1 − 1 ) ⋅ r ∗
S = { a 1 , a 1 + 1 , … , b 1 } ∪ … ∪ { a t , a t + 1 , … , b t } S=\{a_1,a_1+1,\ldots,b_1\}\cup\ldots\cup\{a_t,a_t+1,\ldots,b_t\} S = { a 1 , a 1 + 1 , … , b 1 } ∪ … ∪ { a t , a t + 1 , … , b t } for t ≤ 2 q 2 t\le2q^2 t ≤ 2 q 2 [ a s , b s ] ⊂ S [a_s,b_s]\subset S [ a s , b s ] ⊂ S s ∈ [ t ] s\in[t] s ∈ [ t ]
S ( C 1 ⋅ r ∗ ) = S ( r ∗ + η ) = M ∩ ⋃ s = 1 t { a s , a s + 1 , … , b s , b s + 1 , … , b s + η } , S(C_1\cdot r_*)=S(r_*+\eta)=M\cap\bigcup_{s=1}^{t}\{a_s,a_s+1,\ldots,b_s,b_s+1,\ldots,b_s+\eta\}, S ( C 1 ⋅ r ∗ ) = S ( r ∗ + η ) = M ∩ s = 1 ⋃ t { a s , a s + 1 , … , b s , b s + 1 , … , b s + η } , therefore
F = F ( r ∗ ) = S ( C 1 ⋅ r ∗ ) − S ( r ∗ ) = E ∩ ⋃ s = 1 t { b s + 1 , … , b s + η } . F=F(r_*)=S(C_1\cdot r_*)-S(r_*)=E\cap\bigcup_{s=1}^{t}\{b_s+1,\ldots,b_s+\eta\}. F = F ( r ∗ ) = S ( C 1 ⋅ r ∗ ) − S ( r ∗ ) = E ∩ s = 1 ⋃ t { b s + 1 , … , b s + η } . Consider some agent i ∈ N − W i\in N-W i ∈ N − W α ∈ Q \alpha\in Q α ∈ Q ℓ = m i ( α ) \ell=m_i(\alpha) ℓ = m i ( α ) E E E [ a s , b s ] ⊂ S [a_s,b_s]\subset S [ a s , b s ] ⊂ S S S S ℓ \ell ℓ b s < ℓ b_s<\ell b s < ℓ b s b_s b s 1 ∈ S 0 ⊆ S 1\in S_0\subseteq S 1 ∈ S 0 ⊆ S
By definition of W W W ℓ ∉ F \ell\notin F ℓ ∈ / F ℓ > b s + η \ell>b_s+\eta ℓ > b s + η ℓ ∈ E ∩ { b s + 1 , … , b s + η } ⊆ F \ell\in E\cap\{b_s+1,\ldots,b_s+\eta\}\subseteq F ℓ ∈ E ∩ { b s + 1 , … , b s + η } ⊆ F [ a s , b s ] [a_s,b_s] [ a s , b s ] { b s + 1 , … , b s + η , … , ℓ } ⊆ E \{b_s+1,\ldots,b_s+\eta,\ldots,\ell\}\subseteq E { b s + 1 , … , b s + η , … , ℓ } ⊆ E
{ ℓ − η + 1 , … , ℓ } ⊆ { b s + 1 , … , b s + η , … , ℓ } ⊆ E . \{\ell-\eta+1,\ldots,\ell\}\subseteq\{b_s+1,\ldots,b_s+\eta,\ldots,\ell\}\subseteq E. { ℓ − η + 1 , … , ℓ } ⊆ { b s + 1 , … , b s + η , … , ℓ } ⊆ E . In particular, ℓ − η + 1 ≥ 1 ⟹ ℓ ≥ η \ell-\eta+1\ge1\Longrightarrow\ell\ge\eta ℓ − η + 1 ≥ 1 ⟹ ℓ ≥ η
It remains to bound the number of different breakpoint occurrences in ( ℓ − η , ℓ ] (\ell-\eta,\ell] ( ℓ − η , ℓ ] M ( Q ) \mathcal{M}(Q) M ( Q ) ( ℓ − η , ℓ ] (\ell-\eta,\ell] ( ℓ − η , ℓ ] M ( Q ) \mathcal{M}(Q) M ( Q ) ⌊ n / 2 ⌋ + 1 \lfloor n/2\rfloor+1 ⌊ n /2 ⌋ + 1 M 0 \mathcal{M}_0 M 0 S 0 ∩ ( ℓ − η , ℓ ] S_0\cap(\ell-\eta,\ell] S 0 ∩ ( ℓ − η , ℓ ] ( ℓ − η , ℓ ] (\ell-\eta,\ell] ( ℓ − η , ℓ ] E E E
∣ { ( i , α ) : i ∈ N , α ∈ Q and ℓ − η < m i ( α ) ≤ ℓ } ∣ ≤ ⌊ n / 2 ⌋ ≤ n / 2. |\{(i,\alpha):i\in N,\alpha\in Q\text{ and }\ell-\eta<m_i(\alpha)\le\ell\}|\le\lfloor n/2\rfloor\le n/2. ∣ {( i , α ) : i ∈ N , α ∈ Q and ℓ − η < m i ( α ) ≤ ℓ } ∣ ≤ ⌊ n /2 ⌋ ≤ n /2. Note that all repetitions are counted separately. Breakpoints that fall on the same position cause no issue, as ( ℓ − η , ℓ ] (\ell-\eta,\ell] ( ℓ − η , ℓ ]
Protected prefixes
Given Lemma 4.1, we define the partitions M = E + S M = E + S M = E + S N = U + V + W N = U + V + W N = U + V + W
Definition 4.2. Given the sets W W W S S S E : = M − S E := M - S E := M − S U U U ⌊ ε n / 8 ⌋ \lfloor\varepsilon n/8 \rfloor ⌊ ε n /8 ⌋ i ∈ N − W i \in N - W i ∈ N − W h i h_i h i V : = N − U − W V := N - U - W V := N − U − W
In addition, we will explicitly define the set of protected prefixes L S ⊂ S \mathcal{L}_S \subset S L S ⊂ S A 0 ⊆ A A_0 \subseteq A A 0 ⊆ A U 0 ⊆ U U_0 \subseteq U U 0 ⊆ U 0 ≤ ℓ ≤ m 0 \le\ell\le m 0 ≤ ℓ ≤ m ℓ s ∈ L S \ell_s \in\mathcal{L}_S ℓ s ∈ L S ∣ U 0 ∣ |U_0| ∣ U 0 ∣
Definition 4.3. Decompose set S S S
S = { a 1 , a 1 + 1 , … , b 1 } ∪ … ∪ { a t , a t + 1 , … , b t } , S = \{a_1, a_1 + 1, \ldots, b_1\} \cup\ldots\cup\{a_t, a_t + 1, \ldots, b_t\}, S = { a 1 , a 1 + 1 , … , b 1 } ∪ … ∪ { a t , a t + 1 , … , b t } , where each [ a s , b s ] [a_s, b_s] [ a s , b s ] s ∈ [ t ] s \in[t] s ∈ [ t ] { 1 , … , m } \{1, \ldots, m\} { 1 , … , m } L S ⊂ S \mathcal{L}_S \subset S L S ⊂ S protected prefixes as
L S : = { b s : s ∈ [ t ] } . \mathcal{L}_S := \{b_s : s \in[t]\}. L S := { b s : s ∈ [ t ]} . Note that as t ≤ 2 q 2 t \le2q^2 t ≤ 2 q 2 ∣ L S ∣ ≤ 2 q 2 |\mathcal{L}_S| \le2q^2 ∣ L S ∣ ≤ 2 q 2
Corollary 4.1. Let A ⊆ V A \subseteq V A ⊆ V { π i } i ∈ A \{\pi_i\}_{i \in A} { π i } i ∈ A E E E S S S V V V A 0 ⊆ A A_0 \subseteq A A 0 ⊆ A U 0 ⊆ U U_0 \subseteq U U 0 ⊆ U { π u } u ∈ U 0 \{\pi_u\}_{u \in U_0} { π u } u ∈ U 0 ℓ s ∈ L S \ell_s \in\mathcal{L}_S ℓ s ∈ L S
∑ i ∈ A 0 P i ( π i E , ℓ s ) + ∑ u ∈ U 0 P u ( π u E , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣ . (3) \sum_{i \in A_0} \mathcal{P}_i(\pi_i^E, \ell_s) + \sum_{u \in U_0} \mathcal{P}_u(\pi_u^E, \ell_s) \le|E \cap[\ell_s]|. \tag*{(3)} i ∈ A 0 ∑ P i ( π i E , ℓ s ) + u ∈ U 0 ∑ P u ( π u E , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣. (3) as well as ∣ U 0 ∣ ≤ η / ( 3 q ) |U_0| \le\eta/(3q) ∣ U 0 ∣ ≤ η / ( 3 q ) η \eta η 0 ≤ ℓ ≤ m 0 \le\ell\le m 0 ≤ ℓ ≤ m
∑ i ∈ A 0 P i ( π i E , ℓ ) + ∑ u ∈ U 0 P u ( π u E , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣ . (4) \sum_{i \in A_0} \mathcal{P}_i(\pi_i^E, \ell) + \sum_{u \in U_0} \mathcal{P}_u(\pi_u^E, \ell) \le|E \cap[\ell]|. \tag*{(4)} i ∈ A 0 ∑ P i ( π i E , ℓ ) + u ∈ U 0 ∑ P u ( π u E , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣. (4) Proof. We will prove the statement separately for prefixes from E E E S S S ℓ ∈ S \ell\in S ℓ ∈ S ℓ ∈ L S \ell\in\mathcal{L}_S ℓ ∈ L S L S _{\mathcal{L}_S} L S ℓ ∉ L S \ell\notin\mathcal{L}_S ℓ ∈ / L S ℓ ∈ [ a s , b s ] \ell\in[a_s, b_s] ℓ ∈ [ a s , b s ] [ a s , b s ] ⊂ S [a_s, b_s] \subset S [ a s , b s ] ⊂ S
∀ i ∈ A 0 : P i ( π i E , ℓ ) ≤ P i ( π i E , b s ) and ∀ u ∈ U 0 : P u ( π u E , ℓ ) ≤ P u ( π u E , b s ) . \forall i \in A_0:\ \mathcal{P}_i(\pi_i^E, \ell) \le\mathcal{P}_i(\pi_i^E, b_s)
\quad\text{and}\quad
\forall u \in U_0:\ \mathcal{P}_u(\pi_u^E, \ell) \le\mathcal{P}_u(\pi_u^E, b_s). ∀ i ∈ A 0 : P i ( π i E , ℓ ) ≤ P i ( π i E , b s ) and ∀ u ∈ U 0 : P u ( π u E , ℓ ) ≤ P u ( π u E , b s ) . Since b s ∈ L S b_s \in\mathcal{L}_S b s ∈ L S ( ℓ , b s ] (\ell, b_s] ( ℓ , b s ] S S S
∑ i ∈ A 0 P i ( π i E , ℓ ) + ∑ u ∈ U 0 P u ( π u E , ℓ ) ≤ ∣ E ∩ [ b s ] ∣ = ∣ E ∩ [ ℓ ] ∣ . \sum_{i \in A_0} \mathcal{P}_i(\pi_i^E, \ell) + \sum_{u \in U_0} \mathcal{P}_u(\pi_u^E, \ell) \le|E \cap[b_s]| = |E \cap[\ell]|. i ∈ A 0 ∑ P i ( π i E , ℓ ) + u ∈ U 0 ∑ P u ( π u E , ℓ ) ≤ ∣ E ∩ [ b s ] ∣ = ∣ E ∩ [ ℓ ] ∣. Next, consider ℓ ∈ E \ell\in E ℓ ∈ E ℓ \ell ℓ i ∈ N − W i \in N - W i ∈ N − W { ℓ − η + 1 , ℓ } ⊆ E \{\ell- \eta+ 1, \ell\} \subseteq E { ℓ − η + 1 , ℓ } ⊆ E ( ℓ − η , ℓ ] (\ell- \eta, \ell] ( ℓ − η , ℓ ] n / 2 n/2 n /2
∑ i ∈ A 0 P i ( π i E , ℓ ) ≤ 1 ∣ V ∣ ∑ i ∈ V ∑ j = 1 d P i ( π i , j E , ℓ ) . \sum_{i \in A_0} \mathcal{P}_i(\pi_i^E, \ell)
\le\frac{1}{|V|} \sum_{i \in V} \sum_{j=1}^{d} \mathcal{P}_i(\pi_{i,j}^E, \ell). i ∈ A 0 ∑ P i ( π i E , ℓ ) ≤ ∣ V ∣ 1 i ∈ V ∑ j = 1 ∑ d P i ( π i , j E , ℓ ) . Let V η , ℓ ⊂ V V_{\eta,\ell} \subset V V η , ℓ ⊂ V i ∈ V i \in V i ∈ V ( ℓ − η , ℓ ] (\ell- \eta, \ell] ( ℓ − η , ℓ ] ∣ V η , ℓ ∣ ≤ n / 2 |V_{\eta,\ell}| \le n/2 ∣ V η , ℓ ∣ ≤ n /2 i ∈ V η , ℓ i \in V_{\eta,\ell} i ∈ V η , ℓ
∑ j = 1 d P i ( π i , j E , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣ . \sum_{j=1}^{d} \mathcal{P}_i(\pi_{i,j}^E, \ell) \le|E \cap[\ell]|. j = 1 ∑ d P i ( π i , j E , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣. On the other hand, for every agent i ∈ V − V η , ℓ i \in V - V_{\eta,\ell} i ∈ V − V η , ℓ ( ℓ − η , ℓ ] (\ell-\eta,\ell] ( ℓ − η , ℓ ]
∑ j = 1 d P i ( π i , j E , ℓ ) = ∑ j = 1 d P i ( π i , j E , ℓ − η ) ≤ ∣ E ∩ [ ℓ − η ] ∣ = ∣ E ∩ [ ℓ ] ∣ − η . \sum_{j=1}^{d} \mathcal{P}_{i}(\pi^{E}_{i,j},\ell)=\sum_{j=1}^{d}\mathcal{P}_{i}(\pi^{E}_{i,j},\ell-\eta)\leq|E\cap[\ell-\eta]|=|E\cap[\ell]|-\eta. j = 1 ∑ d P i ( π i , j E , ℓ ) = j = 1 ∑ d P i ( π i , j E , ℓ − η ) ≤ ∣ E ∩ [ ℓ − η ] ∣ = ∣ E ∩ [ ℓ ] ∣ − η . The last equality is due to ( ℓ − η , ℓ ] ⊂ E (\ell-\eta,\ell]\subset E ( ℓ − η , ℓ ] ⊂ E V η , ℓ V_{\eta,\ell} V η , ℓ V − V η , ℓ V-V_{\eta,\ell} V − V η , ℓ
1 ∣ V ∣ ∑ i ∈ V ∑ j = 1 d P i ( π i , j E , ℓ ) ≤ ∣ V ∣ − ∣ V η , ℓ ∣ ∣ V ∣ ⋅ ( ∣ E ∩ [ ℓ ] ∣ − η ) + ∣ V η , ℓ ∣ ∣ V ∣ ⋅ ∣ E ∩ [ ℓ ] ∣ = ∣ E ∩ [ ℓ ] ∣ − η ⋅ ( 1 − ∣ V η , ℓ ∣ ∣ V ∣ ) ≤ ∣ E ∩ [ ℓ ] ∣ − η 3 . \begin{aligned}
\frac{1}{|V|}\sum_{i\in V}\sum_{j=1}^{d}\mathcal{P}_{i}(\pi^{E}_{i,j},\ell)\leq\frac{|V|-|V_{\eta,\ell}|}{|V|}\cdot\left(|E\cap[\ell]|-\eta\right)+\frac{|V_{\eta,\ell}|}{|V|}\cdot|E\cap[\ell]| \\
&=|E\cap[\ell]|-\eta\cdot\left(1-\frac{|V_{\eta,\ell}|}{|V|}\right)\leq|E\cap[\ell]|-\frac{\eta}{3}.
\end{aligned} ∣ V ∣ 1 i ∈ V ∑ j = 1 ∑ d P i ( π i , j E , ℓ ) ≤ ∣ V ∣ ∣ V ∣ − ∣ V η , ℓ ∣ ⋅ ( ∣ E ∩ [ ℓ ] ∣ − η ) + ∣ V ∣ ∣ V η , ℓ ∣ ⋅ ∣ E ∩ [ ℓ ] ∣ = ∣ E ∩ [ ℓ ] ∣ − η ⋅ ( 1 − ∣ V ∣ ∣ V η , ℓ ∣ ) ≤ ∣ E ∩ [ ℓ ] ∣ − 3 η . We used the fact that ∣ V η , ℓ ∣ ≤ n / 2 |V_{\eta,\ell}|\leq n/2 ∣ V η , ℓ ∣ ≤ n /2 ∣ V ∣ ≥ ( 1 − ε / 4 ) n ≥ 3 n / 4 |V|\geq(1-\varepsilon/4)n\geq3n/4 ∣ V ∣ ≥ ( 1 − ε /4 ) n ≥ 3 n /4
∑ u ∈ U 0 P u ( π u E , ℓ ) ≤ ∑ u ∈ U 0 ∣ π u E ∣ ≤ q U 0 , \sum_{u\in U_{0}}\mathcal{P}_{u}(\pi^{E}_{u},\ell)\leq\sum_{u\in U_{0}}|\pi^{E}_{u}|\leq qU_{0}, u ∈ U 0 ∑ P u ( π u E , ℓ ) ≤ u ∈ U 0 ∑ ∣ π u E ∣ ≤ q U 0 , hence under the assumption ∣ U 0 ∣ ≤ η / ( 3 q ) |U_{0}|\leq\eta/(3q) ∣ U 0 ∣ ≤ η / ( 3 q ) ℓ \ell ℓ
∑ i ∈ A 0 P i ( π i E , ℓ ) + ∑ u ∈ U 0 P u ( π u E , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣ . \sum_{i\in A_{0}}\mathcal{P}_{i}(\pi^{E}_{i},\ell)+\sum_{u\in U_{0}}\mathcal{P}_{u}(\pi^{E}_{u},\ell)\leq|E\cap[\ell]|. i ∈ A 0 ∑ P i ( π i E , ℓ ) + u ∈ U 0 ∑ P u ( π u E , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣. Now, consider ℓ ∈ E \ell\in E ℓ ∈ E i ∈ N − W i\in N-W i ∈ N − W ℓ s ∈ L S \ell_{s}\in\mathcal{L}_{S} ℓ s ∈ L S ℓ \ell ℓ 1 ∈ S 1\in S 1 ∈ S ℓ ∗ ∈ ( ℓ s , ℓ ] \ell_{*}\in(\ell_{s},\ell] ℓ ∗ ∈ ( ℓ s , ℓ ] i ∈ N − W i\in N-W i ∈ N − W
∑ i ∈ A 0 P i ( π i E , ℓ ) + ∑ u ∈ U 0 P u ( π u E , ℓ ) = ∑ i ∈ A 0 P i ( π i E , ℓ ∗ ) + ∑ u ∈ U 0 P u ( π u E , ℓ ∗ ) ≤ ∣ E ∩ [ ℓ ∗ ] ∣ ≤ ∣ E ∩ [ ℓ ] ∣ , \sum_{i\in A_{0}}\mathcal{P}_{i}(\pi^{E}_{i},\ell)+\sum_{u\in U_{0}}\mathcal{P}_{u}(\pi^{E}_{u},\ell)=\sum_{i\in A_{0}}\mathcal{P}_{i}(\pi^{E}_{i},\ell_{*})+\sum_{u\in U_{0}}\mathcal{P}_{u}(\pi^{E}_{u},\ell_{*})\leq|E\cap[\ell_{*}]|\leq|E\cap[\ell]|, i ∈ A 0 ∑ P i ( π i E , ℓ ) + u ∈ U 0 ∑ P u ( π u E , ℓ ) = i ∈ A 0 ∑ P i ( π i E , ℓ ∗ ) + u ∈ U 0 ∑ P u ( π u E , ℓ ∗ ) ≤ ∣ E ∩ [ ℓ ∗ ] ∣ ≤ ∣ E ∩ [ ℓ ] ∣ , since prefix load of an agent is constant between consecutive breakpoints. If there are no breakpoints in ( ℓ s , ℓ ] (\ell_{s},\ell] ( ℓ s , ℓ ] ℓ \ell ℓ ℓ s \ell_{s} ℓ s
∑ i ∈ A 0 P i ( π i E , ℓ ) + ∑ u ∈ U 0 P u ( π u E , ℓ ) = ∑ i ∈ A 0 P i ( π i E , ℓ s ) + ∑ u ∈ U 0 P u ( π u E , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣ ≤ ∣ E ∩ [ ℓ ] ∣ , \sum_{i\in A_{0}}\mathcal{P}_{i}(\pi^{E}_{i},\ell)+\sum_{u\in U_{0}}\mathcal{P}_{u}(\pi^{E}_{u},\ell)=\sum_{i\in A_{0}}\mathcal{P}_{i}(\pi^{E}_{i},\ell_{s})+\sum_{u\in U_{0}}\mathcal{P}_{u}(\pi^{E}_{u},\ell_{s})\leq|E\cap[\ell_{s}]|\leq|E\cap[\ell]|, i ∈ A 0 ∑ P i ( π i E , ℓ ) + u ∈ U 0 ∑ P u ( π u E , ℓ ) = i ∈ A 0 ∑ P i ( π i E , ℓ s ) + u ∈ U 0 ∑ P u ( π u E , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣ ≤ ∣ E ∩ [ ℓ ] ∣ , by assumption ( E L S ) (E_{\mathcal{L}_{S}}) ( E L S ) ℓ = 0 \ell=0 ℓ = 0 S S S [ 1 , b 1 ] [1,b_{1}] [ 1 , b 1 ] b 1 ∈ L S b_{1}\in\mathcal{L}_{S} b 1 ∈ L S ∣ E ∩ [ b 1 ] ∣ = 0 |E\cap[b_{1}]|=0 ∣ E ∩ [ b 1 ] ∣ = 0 ( E L S ) (E_{\mathcal{L}_{S}}) ( E L S )
∑ i ∈ A 0 P i ( π i E , 0 ) + ∑ u ∈ U 0 P u ( π u E , 0 ) ≤ ∑ i ∈ A 0 P i ( π i E , b 1 ) + ∑ u ∈ U 0 P u ( π u E , b 1 ) ≤ ∣ E ∩ [ b 1 ] ∣ = 0. \sum_{i\in A_{0}}\mathcal{P}_{i}(\pi^{E}_{i},0)+\sum_{u\in U_{0}}\mathcal{P}_{u}(\pi^{E}_{u},0)\leq\sum_{i\in A_{0}}\mathcal{P}_{i}(\pi^{E}_{i},b_{1})+\sum_{u\in U_{0}}\mathcal{P}_{u}(\pi^{E}_{u},b_{1})\leq|E\cap[b_{1}]|=0. i ∈ A 0 ∑ P i ( π i E , 0 ) + u ∈ U 0 ∑ P u ( π u E , 0 ) ≤ i ∈ A 0 ∑ P i ( π i E , b 1 ) + u ∈ U 0 ∑ P u ( π u E , b 1 ) ≤ ∣ E ∩ [ b 1 ] ∣ = 0. We show how to find sets A 0 ⊆ A A_{0}\subseteq A A 0 ⊆ A U 0 ⊆ U U_{0}\subseteq U U 0 ⊆ U
Supplier and recipients
Following Corollary 4.1, we now have a concrete goal to satisfy conditions (E) and (S): find sets A 0 ⊆ A A_{0}\subseteq A A 0 ⊆ A U 0 ⊆ U U_{0}\subseteq U U 0 ⊆ U ∣ A 0 ∣ + ∣ U 0 ∣ = d |A_{0}|+|U_{0}|=d ∣ A 0 ∣ + ∣ U 0 ∣ = d
∑ i ∈ A 0 P i ( π i E , ℓ s ) + ∑ u ∈ U 0 P u ( π u E , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣ , ∀ ℓ s ∈ L S \sum_{i\in A_{0}}\mathcal{P}_{i}(\pi^{E}_{i},\ell_{s})+\sum_{u\in U_{0}}\mathcal{P}_{u}(\pi^{E}_{u},\ell_{s})\leq|E\cap[\ell_{s}]|,\qquad\forall\ell_{s}\in\mathcal{L}_{S} i ∈ A 0 ∑ P i ( π i E , ℓ s ) + u ∈ U 0 ∑ P u ( π u E , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣ , ∀ ℓ s ∈ L S ∣ U 0 ∣ ≤ η 3 q , |U_{0}|\leq\frac{\eta}{3q}, ∣ U 0 ∣ ≤ 3 q η , ∑ u ∈ U 0 P u ( π u S , ℓ ) ≤ ∣ S ∩ [ ℓ ] ∣ , ∀ 0 ≤ ℓ ≤ m . \sum_{u\in U_{0}}\mathcal{P}_{u}(\pi^{S}_{u},\ell)\leq|S\cap[\ell]|,\qquad\forall0\leq\ell\leq m. u ∈ U 0 ∑ P u ( π u S , ℓ ) ≤ ∣ S ∩ [ ℓ ] ∣ , ∀0 ≤ ℓ ≤ m . Our key idea is the following. Instead of directly choosing agents U 0 ⊆ U U_0 \subseteq U U 0 ⊆ U ( E L S ) (E_{\mathcal{L}_S}) ( E L S ) ( U ) (U) ( U ) ( S ) (S) ( S ) supplier agent u ∗ ∈ U u_* \in U u ∗ ∈ U A A A d d d u ∗ u_* u ∗
More specifically, let { π u ∗ , j : j ∈ [ d ] } \{\pi_{u_*,j}:j\in[d]\} { π u ∗ , j : j ∈ [ d ]} u ∗ ∈ U u_* \in U u ∗ ∈ U J 0 ⊆ [ d ] J_0 \subseteq[d] J 0 ⊆ [ d ] ∣ J 0 ∣ |J_0| ∣ J 0 ∣ u ∗ u_* u ∗ j ∈ J 0 j \in J_0 j ∈ J 0 π u ∗ , j \pi_{u_*,j} π u ∗ , j A 0 ⊆ A A_0 \subseteq A A 0 ⊆ A ∣ A 0 ∣ + ∣ J 0 ∣ = d |A_0|+|J_0|=d ∣ A 0 ∣ + ∣ J 0 ∣ = d ( E L S ) (E_{\mathcal{L}_S}) ( E L S ) ( U ) (U) ( U ) ( S ) (S) ( S ) A 0 A_0 A 0 J 0 J_0 J 0 u ∗ ∈ U u_* \in U u ∗ ∈ U
∑ i ∈ A 0 P i ( π i E , ℓ s ) + ∑ j ∈ J 0 P u ∗ ( π u ∗ , j E , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣ , ∀ ℓ s ∈ L S ; \sum_{i\in A_0}\mathcal{P}_i(\pi_i^E,\ell_s)+\sum_{j\in J_0}\mathcal{P}_{u_*}(\pi_{u_*,j}^E,\ell_s)\leq|E\cap[\ell_s]|,\qquad\forall\ell_s\in\mathcal{L}_S; i ∈ A 0 ∑ P i ( π i E , ℓ s ) + j ∈ J 0 ∑ P u ∗ ( π u ∗ , j E , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣ , ∀ ℓ s ∈ L S ; ∣ J 0 ∣ ≤ η 3 q , |J_0|\leq\frac{\eta}{3q}, ∣ J 0 ∣ ≤ 3 q η , ∑ j ∈ J 0 P u ∗ ( π u ∗ , j S , ℓ ) ≤ ∣ S ∩ [ ℓ ] ∣ , ∀ 0 ≤ ℓ ≤ m . \sum_{j\in J_0}\mathcal{P}_{u_*}(\pi_{u_*,j}^S,\ell)\leq|S\cap[\ell]|,\qquad\forall0\leq\ell\leq m. j ∈ J 0 ∑ P u ∗ ( π u ∗ , j S , ℓ ) ≤ ∣ S ∩ [ ℓ ] ∣ , ∀0 ≤ ℓ ≤ m . Note that condition ( S ∗ ) (S^*) ( S ∗ ) j ∈ J 0 j\in J_0 j ∈ J 0 π u ∗ , j S \pi_{u_*,j}^S π u ∗ , j S B u ∗ , j ∩ S B_{u_*,j}\cap S B u ∗ , j ∩ S B u ∗ , j B_{u_*,j} B u ∗ , j j ∈ J 0 j\in J_0 j ∈ J 0 ℓ \ell ℓ { π u ∗ , j : j ∈ J 0 } \{\pi_{u_*,j}:j\in J_0\} { π u ∗ , j : j ∈ J 0 } ℓ \ell ℓ S ∩ [ ℓ ] S\cap[\ell] S ∩ [ ℓ ] S S S ℓ \ell ℓ ∣ S ∩ [ ℓ ] ∣ |S\cap[\ell]| ∣ S ∩ [ ℓ ] ∣ ( E L S ∗ ) (E_{\mathcal{L}_S}^*) ( E L S ∗ ) ( U ∗ ) (U^*) ( U ∗ )
Given such set A 0 ⊆ A A_0\subseteq A A 0 ⊆ A u ∗ ∈ U u_*\in U u ∗ ∈ U J 0 ⊆ [ d ] J_0\subseteq[d] J 0 ⊆ [ d ] U 0 ⊆ U U_0\subseteq U U 0 ⊆ U ∣ J 0 ∣ |J_0| ∣ J 0 ∣ u ∈ U 0 u\in U_0 u ∈ U 0 supplier’s profile π u ∗ , j \pi_{u_*,j} π u ∗ , j j ∈ J 0 j\in J_0 j ∈ J 0 u ∗ ∈ U u_*\in U u ∗ ∈ U U 0 ⊆ U U_0\subseteq U U 0 ⊆ U J 0 J_0 J 0 u ∗ u_* u ∗ ( E L S ) (E_{\mathcal{L}_S}) ( E L S ) ( U ) (U) ( U ) ( S ) (S) ( S )
Selecting supplier and recipients
Assume ∣ A ∣ < d |A|<d ∣ A ∣ < d u ∗ ∈ U u_*\in U u ∗ ∈ U J 0 ⊆ [ d ] J_0\subseteq[d] J 0 ⊆ [ d ] u ∗ u_* u ∗ A 0 ⊆ A A_0\subseteq A A 0 ⊆ A ∣ A 0 ∣ + ∣ J 0 ∣ = d |A_0|+|J_0|=d ∣ A 0 ∣ + ∣ J 0 ∣ = d ( E L S ∗ ) (E_{\mathcal{L}_S}^*) ( E L S ∗ ) ( U ∗ ) (U^*) ( U ∗ )
Theorem 5.1 (Single Supplier). There exists a global constant C t , q C_{t,q} C t , q For u ∗ ∈ U u_*\in U u ∗ ∈ U { π u ∗ , j : j ∈ [ d ] } \{\pi_{u_*,j}:j\in[d]\} { π u ∗ , j : j ∈ [ d ]} L S ⊆ { 0 , 1 , … , m } \mathcal{L}_S\subseteq\{0,1,\ldots,m\} L S ⊆ { 0 , 1 , … , m } t t t protected prefixes . There exist A 0 ⊆ A A_0\subseteq A A 0 ⊆ A J 0 ⊆ [ d ] J_0\subseteq[d] J 0 ⊆ [ d ] ∣ A 0 ∣ + ∣ J 0 ∣ = d |A_0|+|J_0|=d ∣ A 0 ∣ + ∣ J 0 ∣ = d
∣ J 0 ∣ ≤ C t , q ( d − ∣ A ∣ ) |J_0|\leq C_{t,q}(d-|A|) ∣ J 0 ∣ ≤ C t , q ( d − ∣ A ∣ )
for every protected prefix ℓ s ∈ L S \ell_s\in\mathcal{L}_S ℓ s ∈ L S E E E
∑ i ∈ A 0 P i ( π i E , ℓ s ) + ∑ j ∈ J 0 P u ∗ ( π u ∗ , j E , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣ . \sum_{i\in A_0}\mathcal{P}_i(\pi_i^E,\ell_s)+\sum_{j\in J_0}\mathcal{P}_{u_*}(\pi_{u_*,j}^E,\ell_s)\leq|E\cap[\ell_s]|. i ∈ A 0 ∑ P i ( π i E , ℓ s ) + j ∈ J 0 ∑ P u ∗ ( π u ∗ , j E , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣. We prove Theorem 5.1 in Appendix § 11 (Theorem C.1).
Note that the bound given by Theorem 5.1 on ∣ J 0 ∣ |J_0| ∣ J 0 ∣ C t , q ( d − ∣ A ∣ ) C_{t,q}(d-|A|) C t , q ( d − ∣ A ∣ ) C t , q C_{t,q} C t , q ( U ∗ ) (U^*) ( U ∗ ) ∣ J 0 ∣ ≤ η / ( 3 q ) |J_0|\leq\eta/(3q) ∣ J 0 ∣ ≤ η / ( 3 q ) r 1 , C 1 r_1,C_1 r 1 , C 1 d − ∣ A ∣ ≤ 2 ∣ S ∣ d-|A|\leq2|S| d − ∣ A ∣ ≤ 2∣ S ∣ η ≥ C 1 − 1 2 q 2 ∣ S ∣ \eta\geq\frac{C_1-1}{2q^2}|S| η ≥ 2 q 2 C 1 − 1 ∣ S ∣ ∣ J 0 ∣ |J_0| ∣ J 0 ∣ η \eta η
We will now describe how to replace the J 0 J_0 J 0 u ∗ u_* u ∗ U 0 ⊆ U U_0\subseteq U U 0 ⊆ U j ∈ J 0 j\in J_0 j ∈ J 0 u ( j ) ∈ U u(j)\in U u ( j ) ∈ U supplier’s profile:
∀ j ∈ J 0 : take u ( j ) ∈ U 0 and assign π u ( j ) : = π u ∗ , j . (5) \forall j\in J_0:\ \text{take }u(j)\in U_0\text{ and assign }\pi_{u(j)}:=\pi_{u_*,j}. \tag*{(5)} ∀ j ∈ J 0 : take u ( j ) ∈ U 0 and assign π u ( j ) := π u ∗ , j . (5) This automatically gives
∣ U 0 ∣ = ∣ J 0 ∣ ≤ C t , q ( d − ∣ A ∣ ) , |U_0|=|J_0|\le C_{t,q}(d-|A|), ∣ U 0 ∣ = ∣ J 0 ∣ ≤ C t , q ( d − ∣ A ∣ ) , and we only need to worry about the two following conditions. First, we need to ensure that the prefix load of π u ∗ , j \pi_{u_*,j} π u ∗ , j u ( j ) u(j) u ( j )
∑ i ∈ A 0 P i ( π i E , ℓ s ) + ∑ j ∈ J 0 P u ( j ) ( π u ∗ , j E , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣ , ∀ ℓ s ∈ L S . \sum_{i\in A_0}\mathcal{P}_i(\pi^E_i,\ell_s)+\sum_{j\in J_0}\mathcal{P}_{u(j)}(\pi^E_{u_*,j},\ell_s)\le|E\cap[\ell_s]|,\qquad\forall\ell_s\in\mathcal{L}_S. i ∈ A 0 ∑ P i ( π i E , ℓ s ) + j ∈ J 0 ∑ P u ( j ) ( π u ∗ , j E , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣ , ∀ ℓ s ∈ L S . {#eq:E_{\mathcal{L}_S}}
By Theorem 5.1, this bound is guaranteed to hold if we consider the prefix load of a single agent u ∗ u_* u ∗ u ∗ u_* u ∗ u ( j ) u(j) u ( j ) u ∗ u_* u ∗ u ( j ) u(j) u ( j ) π u ∗ , j \pi_{u_*,j} π u ∗ , j u ( j ) u(j) u ( j ) j ∈ J 0 j\in J_0 j ∈ J 0
Second, when we considered copies J 0 J_0 J 0 u ∗ u_* u ∗ π u ∗ , j \pi_{u_*,j} π u ∗ , j u ∗ u_* u ∗
∑ j ∈ J 0 P u ∗ ( π u ∗ , j S , ℓ ) ≤ ∣ S ∩ [ ℓ ] ∣ , ∀ 0 ≤ ℓ ≤ m . (6) \sum_{j\in J_0}\mathcal{P}_{u_*}(\pi^S_{u_*,j},\ell)\le|S\cap[\ell]|,\qquad\forall0\le\ell\le m. \tag*{(6)} j ∈ J 0 ∑ P u ∗ ( π u ∗ , j S , ℓ ) ≤ ∣ S ∩ [ ℓ ] ∣ , ∀0 ≤ ℓ ≤ m . (6) This does not necessarily hold when we give profiles π u ∗ , j \pi_{u_*,j} π u ∗ , j u ( j ) u(j) u ( j ) u ( j ) u(j) u ( j ) j ∈ J 0 j\in J_0 j ∈ J 0 S S S π u ∗ , j \pi_{u_*,j} π u ∗ , j j ∈ J 0 j\in J_0 j ∈ J 0 u ( j ) u(j) u ( j )
∑ j ∈ J 0 P u ( j ) ( π u ∗ , j S , ℓ ) ≤ ∣ S ∩ [ ℓ ] ∣ , ∀ 0 ≤ ℓ ≤ m . (7) \sum_{j\in J_0}\mathcal{P}_{u(j)}(\pi^S_{u_*,j},\ell)\le|S\cap[\ell]|,\qquad\forall0\le\ell\le m. \tag*{(7)} j ∈ J 0 ∑ P u ( j ) ( π u ∗ , j S , ℓ ) ≤ ∣ S ∩ [ ℓ ] ∣ , ∀0 ≤ ℓ ≤ m . (7) Finally, U 0 U_0 U 0 ∣ J 0 ∣ |J_0| ∣ J 0 ∣ ∣ A 0 ∣ + ∣ U 0 ∣ = ∣ A 0 ∣ + ∣ J 0 ∣ = d |A_0|+|U_0|=|A_0|+|J_0|=d ∣ A 0 ∣ + ∣ U 0 ∣ = ∣ A 0 ∣ + ∣ J 0 ∣ = d u ∗ ∈ U u_*\in U u ∗ ∈ U ∣ J 0 ∣ |J_0| ∣ J 0 ∣ U 0 ⊆ U U_0\subseteq U U 0 ⊆ U
Lemma 5.1 (Supplier and recipients choice). Let L S ⊆ { 0 , 1 , … , m } \mathcal{L}_S\subseteq\{0,1,\ldots,m\} L S ⊆ { 0 , 1 , … , m } t t t U 0 U_0 U 0
∣ U 0 ∣ ≥ ⌊ ∣ U ∣ ( t + ∣ S ∣ + 1 ) q 2 ⌋ |U_0|\ge\left\lfloor\frac{|U|}{(t+|S|+1)^{q^2}}\right\rfloor ∣ U 0 ∣ ≥ ⌊ ( t + ∣ S ∣ + 1 ) q 2 ∣ U ∣ ⌋ with the following property. Let u ∗ ∈ U 0 u_*\in U_0 u ∗ ∈ U 0 J 0 ⊆ [ d ] J_0\subseteq[d] J 0 ⊆ [ d ] ∣ J 0 ∣ ≤ ∣ U 0 ∣ |J_0|\le|U_0| ∣ J 0 ∣ ≤ ∣ U 0 ∣ { π u ∗ , j : j ∈ J 0 } \{\pi_{u_*,j}:j\in J_0\} { π u ∗ , j : j ∈ J 0 } u ∗ u_* u ∗ ∣ J 0 ∣ |J_0| ∣ J 0 ∣ U 0 U_0 U 0 u ( j ) ∈ U 0 u(j)\in U_0 u ( j ) ∈ U 0 π u ∗ , j \pi_{u_*,j} π u ∗ , j
∑ j ∈ J 0 P u ( j ) ( π u ∗ , j E , ℓ s ) = ∑ j ∈ J 0 P u ∗ ( π u ∗ , j E , ℓ s ) , ∀ ℓ s ∈ L S , \sum_{j\in J_0}\mathcal{P}_{u(j)}(\pi^E_{u_*,j},\ell_s)=\sum_{j\in J_0}\mathcal{P}_{u_*}(\pi^E_{u_*,j},\ell_s),\qquad\forall\ell_s\in\mathcal{L}_S, j ∈ J 0 ∑ P u ( j ) ( π u ∗ , j E , ℓ s ) = j ∈ J 0 ∑ P u ∗ ( π u ∗ , j E , ℓ s ) , ∀ ℓ s ∈ L S , ∑ j ∈ J 0 P u ( j ) ( π u ∗ , j S , ℓ ) ≤ ∣ S ∩ [ ℓ ] ∣ , ∀ 0 ≤ ℓ ≤ m . \sum_{j\in J_0}\mathcal{P}_{u(j)}(\pi^S_{u_*,j},\ell)\le|S\cap[\ell]|,\qquad\forall0\le\ell\le m. j ∈ J 0 ∑ P u ( j ) ( π u ∗ , j S , ℓ ) ≤ ∣ S ∩ [ ℓ ] ∣ , ∀0 ≤ ℓ ≤ m . Proof. For each agent u ∈ U u\in U u ∈ U α ∈ Q \alpha\in Q α ∈ Q
whether m u ( α ) ≤ ℓ s m_u(\alpha)\le\ell_s m u ( α ) ≤ ℓ s ℓ s ∈ L S \ell_s\in\mathcal{L}_S ℓ s ∈ L S
1 { m u ( α ) ≤ ℓ 1 } , … , 1 { m u ( α ) ≤ ℓ t } ; \mathbb{1}\{m_u(\alpha)\le\ell_1\},\ldots,\mathbb{1}\{m_u(\alpha)\le\ell_t\}; 1 { m u ( α ) ≤ ℓ 1 } , … , 1 { m u ( α ) ≤ ℓ t } ; which items of S S S α \alpha α u u u
S ∩ [ m u ( α ) ] . S\cap[m_u(\alpha)]. S ∩ [ m u ( α )] . It is easy to see that for every fixed α ∈ Q \alpha\in Q α ∈ Q t + ∣ S ∣ + 1 t+|S|+1 t + ∣ S ∣ + 1 L S \mathcal{L}_S L S S S S { 0 , 1 , … , m } \{0,1,\ldots,m\} { 0 , 1 , … , m } m u ( α ) m_u(\alpha) m u ( α ) m u ( α ) m_u(\alpha) m u ( α ) ℓ s ∈ L S \ell_s \in\mathcal{L}_S ℓ s ∈ L S S S S t + ∣ S ∣ t+|S| t + ∣ S ∣ t + ∣ S ∣ + 1 t+|S|+1 t + ∣ S ∣ + 1 α ∈ Q \alpha\in Q α ∈ Q ∣ Q ∣ ≤ q 2 |Q| \leq q^2 ∣ Q ∣ ≤ q 2 α \alpha α ( t + ∣ S ∣ + 1 ) q 2 (t+|S|+1)^{q^2} ( t + ∣ S ∣ + 1 ) q 2
We partition all agents U U U L S \mathcal{L}_S L S S S S ( t + ∣ S ∣ + 1 ) q 2 (t+|S|+1)^{q^2} ( t + ∣ S ∣ + 1 ) q 2 U 0 U_0 U 0
∣ U 0 ∣ ≥ ⌈ ∣ U ∣ ( t + ∣ S ∣ + 1 ) q 2 ⌉ . |U_0| \geq\left\lceil\frac{|U|}{(t+|S|+1)^{q^2}} \right\rceil. ∣ U 0 ∣ ≥ ⌈ ( t + ∣ S ∣ + 1 ) q 2 ∣ U ∣ ⌉ . For completeness, we show rigorously that all agents within group U 0 U_0 U 0
Pick and arbitrary u ∗ ∈ U 0 u_* \in U_0 u ∗ ∈ U 0 J 0 ⊆ [ d ] J_0 \subseteq[d] J 0 ⊆ [ d ] ∣ J 0 ∣ ≤ ∣ U 0 ∣ |J_0| \leq|U_0| ∣ J 0 ∣ ≤ ∣ U 0 ∣ π u ∗ , j \pi_{u_*,j} π u ∗ , j u ( j ) u(j) u ( j ) j ∈ J 0 j \in J_0 j ∈ J 0 u ∗ u_* u ∗ u ( j ) u(j) u ( j ) α ∈ Q \alpha\in Q α ∈ Q
m u ( j ) ( α ) ≤ ℓ s ⟺ m u ∗ ( α ) ≤ ℓ s , ∀ ℓ s ∈ L S . m_{u(j)}(\alpha) \leq\ell_s \Longleftrightarrow m_{u_*}(\alpha) \leq\ell_s,\qquad\forall\ell_s \in\mathcal{L}_S. m u ( j ) ( α ) ≤ ℓ s ⟺ m u ∗ ( α ) ≤ ℓ s , ∀ ℓ s ∈ L S . It follows that, for every protected prefix ℓ s ∈ L S \ell_s \in\mathcal{L}_S ℓ s ∈ L S
∑ j ∈ J 0 P u ( j ) ( π u ∗ , j E , ℓ s ) = ∑ j ∈ J 0 ∑ α ∈ π u ∗ , j E 1 { m u ( j ) ( α ) ≤ ℓ s } = ∑ j ∈ J 0 ∑ α ∈ π u ∗ , j E 1 { m u ∗ ( α ) ≤ ℓ s } = ∑ j ∈ J 0 P u ∗ ( π u ∗ , j E , ℓ s ) . \sum_{j\in J_0}\mathcal{P}_{u(j)}(\pi^E_{u_*,j},\ell_s)
=
\sum_{j\in J_0}\sum_{\alpha\in\pi^E_{u_*,j}} \mathbf{1}\{m_{u(j)}(\alpha)\leq\ell_s\}
=
\sum_{j\in J_0}\sum_{\alpha\in\pi^E_{u_*,j}} \mathbf{1}\{m_{u_*}(\alpha)\leq\ell_s\}
=
\sum_{j\in J_0}\mathcal{P}_{u_*}(\pi^E_{u_*,j},\ell_s). j ∈ J 0 ∑ P u ( j ) ( π u ∗ , j E , ℓ s ) = j ∈ J 0 ∑ α ∈ π u ∗ , j E ∑ 1 { m u ( j ) ( α ) ≤ ℓ s } = j ∈ J 0 ∑ α ∈ π u ∗ , j E ∑ 1 { m u ∗ ( α ) ≤ ℓ s } = j ∈ J 0 ∑ P u ∗ ( π u ∗ , j E , ℓ s ) . Next, consider some α ∈ Q \alpha\in Q α ∈ Q S ∩ [ m u ( j ) ( α ) ] S \cap[m_{u(j)}(\alpha)] S ∩ [ m u ( j ) ( α )] π u ∗ , j S \pi^S_{u_*,j} π u ∗ , j S e ∈ B u ∗ , j ∩ S e \in B_{u_*,j} \cap S e ∈ B u ∗ , j ∩ S v u ∗ ( e ) ≥ α v_{u_*}(e) \geq\alpha v u ∗ ( e ) ≥ α e ∈ S ∩ [ m u ∗ ( α ) ] e \in S \cap[m_{u_*}(\alpha)] e ∈ S ∩ [ m u ∗ ( α )] e ∈ S ∩ [ m u ( j ) ( α ) ] e \in S \cap[m_{u(j)}(\alpha)] e ∈ S ∩ [ m u ( j ) ( α )] e ∈ B u ∗ , j ∩ S e \in B_{u_*,j} \cap S e ∈ B u ∗ , j ∩ S u ( j ) u(j) u ( j ) π u ( j ) S = π u ∗ , j S \pi^S_{u(j)}=\pi^S_{u_*,j} π u ( j ) S = π u ∗ , j S u ( j ) u(j) u ( j ) j ∈ J 0 j \in J_0 j ∈ J 0 S S S P u ( j ) ( π u ∗ , j S , ℓ ) \mathcal{P}_{u(j)}(\pi^S_{u_*,j},\ell) P u ( j ) ( π u ∗ , j S , ℓ ) S ∩ [ ℓ ] S \cap[\ell] S ∩ [ ℓ ]
∑ j ∈ J 0 P u ( j ) ( π u ∗ , j S , ℓ ) ≤ ∣ S ∩ [ ℓ ] ∣ , ∀ 0 ≤ ℓ ≤ m . \sum_{j\in J_0}\mathcal{P}_{u(j)}(\pi^S_{u_*,j},\ell)\leq|S\cap[\ell]|,\qquad\forall0\leq\ell\leq m. j ∈ J 0 ∑ P u ( j ) ( π u ∗ , j S , ℓ ) ≤ ∣ S ∩ [ ℓ ] ∣ , ∀0 ≤ ℓ ≤ m . □ \square □
Remark 5.1. The denominator in Lemma 5.1 can be improved to O ( ( t ∣ S ∣ ) q ) O((t|S|)^{q}) O (( t ∣ S ∣ ) q ) u ∈ U u\in U u ∈ U u ∈ U u\in U u ∈ U π u , j \pi_{u,j} π u , j j ∈ [ d ] j\in[d] j ∈ [ d ] E E E S S S E E E S S S S S S U U U G = O ( ( t ∣ S ∣ ) q ) G=O((t|S|)^q) G = O (( t ∣ S ∣ ) q ) ∣ U ∣ / G |U|/G ∣ U ∣/ G u ∗ u_* u ∗ j ∈ J 0 j\in J_0 j ∈ J 0 ( u ∗ , j ) (u_*,j) ( u ∗ , j ) u ( j ) u(j) u ( j ) n ε , q n_{\varepsilon,q} n ε , q § 12 becomes 2 O ( q 5 / ε ) log q 2^{O(q^5/\varepsilon)\log q} 2 O ( q 5 / ε ) l o g q 2 O ( q 6 / ε ) log q 2^{O(q^6/\varepsilon)\log q} 2 O ( q 6 / ε ) l o g q
It follows that for sufficiently large number of agents n n n U 0 ⊆ U U_0\subseteq U U 0 ⊆ U u ∗ ∈ U 0 u_*\in U_0 u ∗ ∈ U 0 J 0 J_0 J 0 U 0 U_0 U 0 ( E L S ) (E_{\mathcal{L}_S}) ( E L S ) ( U ) (U) ( U ) ( S ) (S) ( S ) A 0 ⊆ A A_0\subseteq A A 0 ⊆ A
Corollary 5.1. For every fixed ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 ) q ≥ 3 q \ge3 q ≥ 3 n ε , q n_{\varepsilon,q} n ε , q n ≥ n ε , q n \ge n_{\varepsilon,q} n ≥ n ε , q γ = min ( ε , 1 / ( 2 q ) ) \gamma= \min(\varepsilon,1/(2q)) γ = min ( ε , 1/ ( 2 q )) λ = ( 1 − 1 / ( 2 q ) ) − 1 \lambda= (1-1/(2q))^{-1} λ = ( 1 − 1/ ( 2 q ) ) − 1 N N N
T ‾ : = 1 n ∑ i ∈ N T i < ρ q , n γ , \overline{T}:=\frac{1}{n}\sum_{i\in N}T_i<\frac{\rho_{q,n}}{\gamma}, T := n 1 i ∈ N ∑ T i < γ ρ q , n , where
ρ q , n : = 8 + 4 log λ ( q 3 d 2 ( d + 2 ) n ) + 8 q log ( ( d + 2 ) n ) . \rho_{q,n}:=8+4\log_{\lambda}\left(q^{3}d^{2}(d+2)n\right)+8q\log\left((d+2)n\right). ρ q , n := 8 + 4 log λ ( q 3 d 2 ( d + 2 ) n ) + 8 q log ( ( d + 2 ) n ) . Let A ⊆ V A\subseteq V A ⊆ V π i \pi_i π i i ∈ A i\in A i ∈ A ∣ A ∣ < d |A|<d ∣ A ∣ < d U 0 ⊆ U U_0\subseteq U U 0 ⊆ U u ∗ ∈ U 0 u_*\in U_0 u ∗ ∈ U 0 { π u ∗ , j : j ∈ [ d ] } \{\pi_{u_*,j}:j\in[d]\} { π u ∗ , j : j ∈ [ d ]} A 0 ⊆ A A_0\subseteq A A 0 ⊆ A J 0 ⊆ [ d ] J_0\subseteq[d] J 0 ⊆ [ d ] ∣ J 0 ∣ ≤ ∣ U 0 ∣ |J_0|\leq|U_0| ∣ J 0 ∣ ≤ ∣ U 0 ∣ ∣ A 0 ∣ + ∣ J 0 ∣ = d |A_0|+|J_0|=d ∣ A 0 ∣ + ∣ J 0 ∣ = d j ∈ J 0 j\in J_0 j ∈ J 0 π u ∗ , j \pi_{u_*,j} π u ∗ , j u ( j ) ∈ U 0 u(j)\in U_0 u ( j ) ∈ U 0
∑ i ∈ A 0 P i ( π i , ℓ ) + ∑ j ∈ J 0 P u ( j ) ( π u ∗ , j , ℓ ) ≤ ℓ , ∀ 0 ≤ ℓ ≤ m . \sum_{i\in A_0}\mathcal{P}_i(\pi_i,\ell)+\sum_{j\in J_0}\mathcal{P}_{u(j)}(\pi_{u_*,j},\ell)\leq\ell,\qquad\forall0\leq\ell\leq m. i ∈ A 0 ∑ P i ( π i , ℓ ) + j ∈ J 0 ∑ P u ( j ) ( π u ∗ , j , ℓ ) ≤ ℓ , ∀0 ≤ ℓ ≤ m . Proof. We will first bound the number d − ∣ A ∣ d-|A| d − ∣ A ∣ ρ A = 8 + 16 q log n \rho_A=8+16q\log n ρ A = 8 + 16 q log n ∣ V ∣ ≥ ( 1 − ε / 4 ) n ≥ 3 n / 4 |V|\geq(1-\varepsilon/4)n\geq3n/4 ∣ V ∣ ≥ ( 1 − ε /4 ) n ≥ 3 n /4
d − ∣ A ∣ ≤ ∣ S ∣ + q ( q − 1 ) ∣ V ∣ ∑ i ∈ V T i + ρ A ≤ ∣ S ∣ + 2 q ( q − 1 ) T ‾ + ρ A ≤ ∣ S ∣ + 2 q ( q − 1 ) γ ρ q , n + ρ A . d-|A|\leq|S|+\frac{q(q-1)}{|V|}\sum_{i\in V}T_i+\rho_A\leq|S|+2q(q-1)\overline{T}+\rho_A\leq|S|+\frac{2q(q-1)}{\gamma}\rho_{q,n}+\rho_A. d − ∣ A ∣ ≤ ∣ S ∣ + ∣ V ∣ q ( q − 1 ) i ∈ V ∑ T i + ρ A ≤ ∣ S ∣ + 2 q ( q − 1 ) T + ρ A ≤ ∣ S ∣ + γ 2 q ( q − 1 ) ρ q , n + ρ A . The number of protected prefixes ∣ L S ∣ |\mathcal{L}_S| ∣ L S ∣ t ≤ 2 q 2 t\leq2q^2 t ≤ 2 q 2
C q : = max { 1 , max 1 ≤ t ≤ 2 q 2 C t , q } , C_q:=\max\left\{1,\max_{1\leq t\leq2q^2}C_{t,q}\right\}, C q := max { 1 , 1 ≤ t ≤ 2 q 2 max C t , q } , where each C t , q C_{t,q} C t , q
C 1 : = 1 + ⌈ 12 q 3 C q ⌉ and r 1 : = 1 + ⌈ ( 2 q ( q − 1 ) γ + 2 ) ρ q , n ⌉ . C_1:=1+\left\lceil12q^3C_q\right\rceil\qquad\text{and}\qquad r_1:=1+\left\lceil\left(\frac{2q(q-1)}{\gamma}+2\right)\rho_{q,n}\right\rceil. C 1 := 1 + ⌈ 12 q 3 C q ⌉ and r 1 := 1 + ⌈ ( γ 2 q ( q − 1 ) + 2 ) ρ q , n ⌉ . Using ρ A ≤ 2 ρ q , n \rho_A\leq2\rho_{q,n} ρ A ≤ 2 ρ q , n
d − ∣ A ∣ ≤ ∣ S ∣ + 2 q ( q − 1 ) γ ρ q , n + ρ A ≤ ∣ S ∣ + r 1 − 1. d-|A|\leq|S|+\frac{2q(q-1)}{\gamma}\rho_{q,n}+\rho_A\leq|S|+r_1-1. d − ∣ A ∣ ≤ ∣ S ∣ + γ 2 q ( q − 1 ) ρ q , n + ρ A ≤ ∣ S ∣ + r 1 − 1. By Lemma 4.1, ∣ S ∣ ≥ r 1 |S|\geq r_1 ∣ S ∣ ≥ r 1 d − ∣ A ∣ ≤ 2 ∣ S ∣ d-|A|\leq2|S| d − ∣ A ∣ ≤ 2∣ S ∣
η ≥ C 1 − 1 2 q 2 ∣ S ∣ , ⟹ C q ( d − ∣ A ∣ ) ≤ 2 C q ∣ S ∣ ≤ C 1 − 1 6 q 3 ∣ S ∣ ≤ η 3 q . \eta\geq\frac{C_1-1}{2q^2}|S|,\qquad\Longrightarrow\qquad C_q(d-|A|)\leq2C_q|S|\leq\frac{C_1-1}{6q^3}|S|\leq\frac{\eta}{3q}. η ≥ 2 q 2 C 1 − 1 ∣ S ∣ , ⟹ C q ( d − ∣ A ∣ ) ≤ 2 C q ∣ S ∣ ≤ 6 q 3 C 1 − 1 ∣ S ∣ ≤ 3 q η . We will now find large enough set of recipients U 0 U_0 U 0 ∣ J 0 ∣ ≤ C q ( d − ∣ A ∣ ) |J_0|\leq C_q(d-|A|) ∣ J 0 ∣ ≤ C q ( d − ∣ A ∣ ) C 1 = O q ( 1 ) C_1=O_q(1) C 1 = O q ( 1 ) r 1 = O ε , q ( log n ) r_1=O_{\varepsilon,q}(\log n) r 1 = O ε , q ( log n ) ∣ S ∣ = O ε , q ( log n ) |S|=O_{\varepsilon,q}(\log n) ∣ S ∣ = O ε , q ( log n ) d − ∣ A ∣ = O ε , q ( log n ) d-|A|=O_{\varepsilon,q}(\log n) d − ∣ A ∣ = O ε , q ( log n ) C ∗ = C ∗ ( ε , q ) ≥ 1 C_*=C_*(\varepsilon,q)\geq1 C ∗ = C ∗ ( ε , q ) ≥ 1
t + ∣ S ∣ + 1 ≤ C ∗ log n , and C q ( d − ∣ A ∣ ) ≤ C ∗ log n . t+|S|+1\leq C_*\log n,\qquad\text{and}\qquad C_q(d-|A|)\leq C_*\log n. t + ∣ S ∣ + 1 ≤ C ∗ log n , and C q ( d − ∣ A ∣ ) ≤ C ∗ log n . Choose n ε , q ≥ 16 / ε n_{\varepsilon,q}\geq16/\varepsilon n ε , q ≥ 16/ ε n ≥ n ε , q n\geq n_{\varepsilon,q} n ≥ n ε , q
n ( log n ) q 2 + 1 ≥ 16 C ∗ q 2 + 1 ε . \frac{n}{(\log n)^{q^2+1}}\geq\frac{16C_*^{q^2+1}}{\varepsilon}. ( log n ) q 2 + 1 n ≥ ε 16 C ∗ q 2 + 1 . Next, by definition, ∣ U ∣ = ⌊ ε n / 8 ⌋ ≥ ε n / 16 |U|=\lfloor\varepsilon n/8\rfloor\geq\varepsilon n/16 ∣ U ∣ = ⌊ ε n /8 ⌋ ≥ ε n /16 U 0 ⊆ U U_0\subseteq U U 0 ⊆ U
∣ U 0 ∣ ≥ ∣ U ∣ ( t + ∣ S ∣ + 1 ) q 2 ≥ ε n 16 ( C ∗ log n ) q 2 ≥ C ∗ log n ≥ C q ( d − ∣ A ∣ ) . |U_0|\geq\frac{|U|}{(t+|S|+1)^{q^2}}\geq\frac{\varepsilon n}{16(C_*\log n)^{q^2}}\geq C_*\log n\geq C_q(d-|A|). ∣ U 0 ∣ ≥ ( t + ∣ S ∣ + 1 ) q 2 ∣ U ∣ ≥ 16 ( C ∗ log n ) q 2 ε n ≥ C ∗ log n ≥ C q ( d − ∣ A ∣ ) . Fix an arbitrary supplier u ∗ ∈ U 0 u_* \in U_0 u ∗ ∈ U 0 A A A u ∗ u_* u ∗ A 0 ⊆ A A_0 \subseteq A A 0 ⊆ A J 0 ⊆ [ d ] J_0 \subseteq[d] J 0 ⊆ [ d ]
∣ A 0 ∣ + ∣ J 0 ∣ = d , and ∣ J 0 ∣ ≤ C q ( d − ∣ A ∣ ) ≤ min { ∣ U 0 ∣ , η / ( 3 q ) } , |A_0| + |J_0| = d, \qquad\text{and} \qquad|J_0| \le C_q(d-|A|) \le\min\{|U_0|,\eta/(3q)\}, ∣ A 0 ∣ + ∣ J 0 ∣ = d , and ∣ J 0 ∣ ≤ C q ( d − ∣ A ∣ ) ≤ min { ∣ U 0 ∣ , η / ( 3 q )} , where for every protected prefix ℓ s ∈ L S \ell_s \in\mathcal{L}_S ℓ s ∈ L S
∑ i ∈ A 0 P i ( π i E , ℓ s ) + ∑ j ∈ J 0 P u ∗ ( π u ∗ , j E , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣ . \sum_{i\in A_0} \mathcal{P}_i(\pi_i^E,\ell_s) + \sum_{j\in J_0} \mathcal{P}_{u_*}(\pi_{u_*,j}^E,\ell_s) \le|E \cap[\ell_s]|. i ∈ A 0 ∑ P i ( π i E , ℓ s ) + j ∈ J 0 ∑ P u ∗ ( π u ∗ , j E , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣. Since ∣ J 0 ∣ ≤ ∣ U 0 ∣ |J_0| \le|U_0| ∣ J 0 ∣ ≤ ∣ U 0 ∣ π u ∗ , j \pi_{u_*,j} π u ∗ , j j ∈ J 0 j \in J_0 j ∈ J 0 u ( j ) ∈ U 0 u(j) \in U_0 u ( j ) ∈ U 0 π u ( j ) = π u ∗ , j \pi_{u(j)}=\pi_{u_*,j} π u ( j ) = π u ∗ , j u ( j ) u(j) u ( j ) E E E S S S
∑ i ∈ A 0 P i ( π i E , ℓ s ) + ∑ j ∈ J 0 P u ( j ) ( π u ∗ , j E , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣ , ∀ ℓ s ∈ L S , \sum_{i\in A_0} \mathcal{P}_i(\pi_i^E,\ell_s) + \sum_{j\in J_0} \mathcal{P}_{u(j)}(\pi_{u_*,j}^E,\ell_s) \le|E \cap[\ell_s]|, \qquad\forall\ell_s \in\mathcal{L}_S, i ∈ A 0 ∑ P i ( π i E , ℓ s ) + j ∈ J 0 ∑ P u ( j ) ( π u ∗ , j E , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣ , ∀ ℓ s ∈ L S , ∑ j ∈ J 0 P u ( j ) ( π u ∗ , j S , ℓ ) ≤ ∣ S ∩ [ ℓ ] ∣ , ∀ 0 ≤ ℓ ≤ m . \sum_{j\in J_0} \mathcal{P}_{u(j)}(\pi_{u_*,j}^S,\ell) \le|S \cap[\ell]|, \qquad\forall0 \le\ell\le m. j ∈ J 0 ∑ P u ( j ) ( π u ∗ , j S , ℓ ) ≤ ∣ S ∩ [ ℓ ] ∣ , ∀0 ≤ ℓ ≤ m . Let U 0 ( J 0 ) : = { u ( j ) : j ∈ J 0 } U_0(J_0):=\{u(j):j\in J_0\} U 0 ( J 0 ) := { u ( j ) : j ∈ J 0 } ∣ U 0 ( J 0 ) ∣ = ∣ J 0 ∣ ≤ η / ( 3 q ) |U_0(J_0)|=|J_0|\le\eta/(3q) ∣ U 0 ( J 0 ) ∣ = ∣ J 0 ∣ ≤ η / ( 3 q ) A 0 A_0 A 0 U 0 ( J 0 ) U_0(J_0) U 0 ( J 0 ) 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m
∑ i ∈ A 0 P i ( π i E , ℓ ) + ∑ j ∈ J 0 P u ( j ) ( π u ∗ , j E , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣ . \sum_{i\in A_0} \mathcal{P}_i(\pi_i^E,\ell) + \sum_{j\in J_0} \mathcal{P}_{u(j)}(\pi_{u_*,j}^E,\ell) \le|E\cap[\ell]|. i ∈ A 0 ∑ P i ( π i E , ℓ ) + j ∈ J 0 ∑ P u ( j ) ( π u ∗ , j E , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣. Since for every i ∈ A 0 i\in A_0 i ∈ A 0 π i E = π i \pi_i^E=\pi_i π i E = π i j ∈ J 0 j\in J_0 j ∈ J 0 π u ∗ , j E ⊔ π u ∗ , j S = π u ∗ , j \pi_{u_*,j}^E \mathbin{\mathop{\sqcup}} \pi_{u_*,j}^S=\pi_{u_*,j} π u ∗ , j E ⊔ π u ∗ , j S = π u ∗ , j E E E S S S
∑ i ∈ A 0 P i ( π i , ℓ ) + ∑ j ∈ J 0 P u ( j ) ( π u ∗ , j , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣ + ∣ S ∩ [ ℓ ] ∣ = ℓ \sum_{i\in A_0} \mathcal{P}_i(\pi_i,\ell) + \sum_{j\in J_0} \mathcal{P}_{u(j)}(\pi_{u_*,j},\ell) \le|E\cap[\ell]|+|S\cap[\ell]|=\ell i ∈ A 0 ∑ P i ( π i , ℓ ) + j ∈ J 0 ∑ P u ( j ) ( π u ∗ , j , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣ + ∣ S ∩ [ ℓ ] ∣ = ℓ for every 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m ∣ A 0 ∣ + ∣ U ( J 0 ) ∣ = ∣ A 0 ∣ + ∣ J 0 ∣ = d |A_0|+|U(J_0)|=|A_0|+|J_0|=d ∣ A 0 ∣ + ∣ U ( J 0 ) ∣ = ∣ A 0 ∣ + ∣ J 0 ∣ = d
Common tail completion
For sets U 0 ⊆ U U_0\subseteq U U 0 ⊆ U J 0 ⊆ [ d ] J_0\subseteq[d] J 0 ⊆ [ d ] u ∗ ∈ U 0 u_*\in U_0 u ∗ ∈ U 0 U 0 ( J 0 ) U_0(J_0) U 0 ( J 0 ) u ( j ) ∈ U 0 u(j)\in U_0 u ( j ) ∈ U 0 j ∈ J 0 j\in J_0 j ∈ J 0 u ( j ) u(j) u ( j ) π u ∗ , j \pi_{u_*,j} π u ∗ , j d d d I = A 0 + U ( J 0 ) I=A_0+U(J_0) I = A 0 + U ( J 0 ) Π I = { π i : i ∈ A 0 } + { π u ∗ , j : j ∈ J 0 } \Pi_I=\{\pi_i:i\in A_0\}+\{\pi_{u_*,j}:j\in J_0\} Π I = { π i : i ∈ A 0 } + { π u ∗ , j : j ∈ J 0 }
∑ i ∈ A 0 P i ( π i , ℓ ) + ∑ j ∈ J 0 P u ( j ) ( π u ∗ , j , ℓ ) ≤ ℓ , ∀ 0 ≤ ℓ ≤ m . \sum_{i\in A_0}\mathcal{P}_i(\pi_i,\ell)+\sum_{j\in J_0}\mathcal{P}_{u(j)}(\pi_{u_*,j},\ell)\le\ell,\qquad\forall0\le\ell\le m. i ∈ A 0 ∑ P i ( π i , ℓ ) + j ∈ J 0 ∑ P u ( j ) ( π u ∗ , j , ℓ ) ≤ ℓ , ∀0 ≤ ℓ ≤ m . So, by Lemma 2.2, we can simultaneously satisfy all demands in Π I \Pi_I Π I M M M i ∈ I i\in I i ∈ I Δ ( π i ) = 0 \Delta(\pi_i)=0 Δ ( π i ) = 0 i i i 1 − 1 / q 1-1/q 1 − 1/ q
This may not be the case: our construction guarantees only for agents i ∈ A 0 i\in A_0 i ∈ A 0 Δ ( π i ) = 0 \Delta(\pi_i)=0 Δ ( π i ) = 0 π u ∗ , j \pi_{u_*,j} π u ∗ , j ℓ \ell ℓ
∑ i ∈ I Δ ( π i ) T i ( ℓ − h i ) + . \sum_{i\in I}\frac{\Delta(\pi_i)}{T_i}(\ell-h_i)_+. i ∈ I ∑ T i Δ ( π i ) ( ℓ − h i ) + . Even if for some Δ ( π i ) > 0 \Delta(\pi_i)>0 Δ ( π i ) > 0 T i T_i T i Δ ( π i ) T i \frac{\Delta(\pi_i)}{T_i} T i Δ ( π i ) Π I \Pi_I Π I I = A 0 + U 0 ( J 0 ) I = A_0 + U_0(J_0) I = A 0 + U 0 ( J 0 ) Π I \Pi_I Π I
In reality, the choice of the original set U U U U 0 ( J 0 ) U_0(J_0) U 0 ( J 0 ) I = A 0 + U 0 ( J 0 ) I = A_0 + U_0(J_0) I = A 0 + U 0 ( J 0 )
Corollary 5.2. All demands in Π I = { π i : i ∈ A 0 } + { π u ∗ , j : j ∈ J 0 } \Pi_I = \{\pi_i : i \in A_0\} + \{\pi_{u_*,j} : j \in J_0\} Π I = { π i : i ∈ A 0 } + { π u ∗ , j : j ∈ J 0 } can be simultaneously satisfied by distinct items in { 1 , … , h ∗ } \{1,\ldots,h_*\} { 1 , … , h ∗ } , where h ∗ h_* h ∗ is the head threshold of the supplier u ∗ u_* u ∗ Furthermore, if T ∗ T_* T ∗ is the total tail value of u ∗ u_* u ∗ it holds
∑ i ∈ A 0 Δ ( π i ) + ∑ j ∈ J 0 Δ ( π u ∗ , j ) ≤ T ∗ . \sum_{i\in A_0}\Delta(\pi_i)+\sum_{j\in J_0}\Delta(\pi_{u_*,j})\le T_*. i ∈ A 0 ∑ Δ ( π i ) + j ∈ J 0 ∑ Δ ( π u ∗ , j ) ≤ T ∗ . Proof. The bound on the total deficit follows by recalling that Δ ( π i ) = 0 \Delta(\pi_i) = 0 Δ ( π i ) = 0 i ∈ A 0 i \in A_0 i ∈ A 0 u ∗ u_* u ∗
Now, consider the proof of Corollary 5.1. After applying Theorem 5.1 to agent u ∗ ∈ U 0 u_* \in U_0 u ∗ ∈ U 0 A A A J 0 ⊆ [ d ] J_0 \subseteq[d] J 0 ⊆ [ d ] u ∗ u_* u ∗ ( E L S ∗ ) (\mathrm{ELS}_*) ( ELS ∗ )
∑ i ∈ A 0 P i ( π i E , ℓ s ) + ∑ j ∈ J 0 P u ∗ ( π u ∗ , j E , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣ , ∀ ℓ s ∈ L S ; (8) \sum_{i\in A_0}\mathcal{P}_i(\pi_i^E,\ell_s)+\sum_{j\in J_0}\mathcal{P}_{u_*}(\pi_{u_*,j}^E,\ell_s)\le|E\cap[\ell_s]|,\qquad\forall\ell_s\in\mathcal{L}_S; \tag*{(8)} i ∈ A 0 ∑ P i ( π i E , ℓ s ) + j ∈ J 0 ∑ P u ∗ ( π u ∗ , j E , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣ , ∀ ℓ s ∈ L S ; (8) Furthermore, Theorem 5.1 guarantees ∣ J 0 ∣ ≤ C q ( d − ∣ A ∣ ) |J_0| \le C_q(d-|A|) ∣ J 0 ∣ ≤ C q ( d − ∣ A ∣ ) C q , C 1 , r 1 C_q,C_1,r_1 C q , C 1 , r 1 C q ( d − ∣ A ∣ ) ≤ η / ( 3 q ) C_q(d-|A|) \le\eta/(3q) C q ( d − ∣ A ∣ ) ≤ η / ( 3 q ) A 0 A_0 A 0 U ∗ U_* U ∗ ∣ J 0 ∣ |J_0| ∣ J 0 ∣ u ∗ u_* u ∗ π u ∗ , j \pi_{u_*,j} π u ∗ , j j ∈ J 0 j \in J_0 j ∈ J 0 0 ≤ ℓ ≤ m 0 \le\ell\le m 0 ≤ ℓ ≤ m
∑ i ∈ A 0 P i ( π i E , ℓ ) + ∑ j ∈ J 0 P u ∗ ( π u ∗ , j E , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣ ; (9) \sum_{i\in A_0}\mathcal{P}_i(\pi_i^E,\ell)+\sum_{j\in J_0}\mathcal{P}_{u_*}(\pi_{u_*,j}^E,\ell)\le|E\cap[\ell]|; \tag*{(9)} i ∈ A 0 ∑ P i ( π i E , ℓ ) + j ∈ J 0 ∑ P u ∗ ( π u ∗ , j E , ℓ ) ≤ ∣ E ∩ [ ℓ ] ∣ ; (9) ∑ j ∈ J 0 P u ∗ ( π u ∗ , j S , ℓ ) ≤ ∣ S ∩ [ ℓ ] ∣ . (10) \sum_{j\in J_0}\mathcal{P}_{u_*}(\pi_{u_*,j}^S,\ell)\le|S\cap[\ell]|. \tag*{(10)} j ∈ J 0 ∑ P u ∗ ( π u ∗ , j S , ℓ ) ≤ ∣ S ∩ [ ℓ ] ∣. (10) Constraints ( S ∗ ) (S_*) ( S ∗ ) π u ∗ , j \pi_{u_*,j} π u ∗ , j u ∗ u_* u ∗
Let h ∗ : = h u ∗ h_* := h_{u_*} h ∗ := h u ∗ U U U ⌊ ε n / 8 ⌋ \lfloor\varepsilon n/8\rfloor ⌊ ε n /8 ⌋ N − W = U + V N-W=U+V N − W = U + V h u h_u h u u ∗ ∈ U u_* \in U u ∗ ∈ U i ∈ A 0 i \in A_0 i ∈ A 0 h i ≤ h ∗ h_i \le h_* h i ≤ h ∗ α ∈ Q \alpha\in Q α ∈ Q m i ( α ) ≤ h ∗ m_i(\alpha) \le h_* m i ( α ) ≤ h ∗ u ∗ u_* u ∗ h ∗ h_* h ∗
∑ i ∈ A 0 ∣ π i E ∣ + ∑ j ∈ J 0 ∣ π u ∗ , j E ∣ = ∑ i ∈ A 0 P i ( π i E , h ∗ ) + ∑ j ∈ J 0 P u ∗ ( π u ∗ , j E , h ∗ ) ≤ ∣ E ∩ [ h ∗ ] ∣ . \sum_{i\in A_0}|\pi_i^E|+\sum_{j\in J_0}|\pi_{u_*,j}^E|
=\sum_{i\in A_0}\mathcal{P}_i(\pi_i^E,h_*)+\sum_{j\in J_0}\mathcal{P}_{u_*}(\pi_{u_*,j}^E,h_*)\le|E\cap[h_*]|. i ∈ A 0 ∑ ∣ π i E ∣ + j ∈ J 0 ∑ ∣ π u ∗ , j E ∣ = i ∈ A 0 ∑ P i ( π i E , h ∗ ) + j ∈ J 0 ∑ P u ∗ ( π u ∗ , j E , h ∗ ) ≤ ∣ E ∩ [ h ∗ ] ∣. as well as
∑ j ∈ J 0 ∣ π u ∗ , j S ∣ = ∑ j ∈ J 0 P u ∗ ( π u ∗ , j S , h ∗ ) ≤ ∣ S ∩ [ h ∗ ] ∣ . \sum_{j\in J_0}|\pi_{u_*,j}^S|
=\sum_{j\in J_0}\mathcal{P}_{u_*}(\pi_{u_*,j}^S,h_*)\le|S\cap[h_*]|. j ∈ J 0 ∑ ∣ π u ∗ , j S ∣ = j ∈ J 0 ∑ P u ∗ ( π u ∗ , j S , h ∗ ) ≤ ∣ S ∩ [ h ∗ ] ∣. Hence, the total number of demands in Π I \Pi_I Π I
K : = ∑ i ∈ A 0 ∣ π i ∣ + ∑ j ∈ J 0 ∣ π u ∗ , j ∣ ≤ ∣ E ∩ [ h ∗ ] ∣ + ∣ S ∩ [ h ∗ ] ∣ = h ∗ . K:=\sum_{i\in A_0}|\pi_i|+\sum_{j\in J_0}|\pi_{u_*,j}|\le|E\cap[h_*]|+|S\cap[h_*]|=h_*. K := i ∈ A 0 ∑ ∣ π i ∣ + j ∈ J 0 ∑ ∣ π u ∗ , j ∣ ≤ ∣ E ∩ [ h ∗ ] ∣ + ∣ S ∩ [ h ∗ ] ∣ = h ∗ . When assigning profiles π u ∗ , j \pi_{u_*,j} π u ∗ , j u ( j ) u(j) u ( j ) j ∈ J 0 j \in J_0 j ∈ J 0 Π I \Pi_I Π I u ( j ) u(j) u ( j ) j ∈ J 0 j \in J_0 j ∈ J 0
∑ i ∈ A 0 P i ( π i , ℓ ) + ∑ j ∈ J 0 P u ( j ) ( π u ∗ , j , ℓ ) ≤ ℓ , ∀ 0 ≤ ℓ ≤ m . \sum_{i\in A_0}\mathcal{P}_i(\pi_i,\ell)+\sum_{j\in J_0}\mathcal{P}_{u(j)}(\pi_{u_*,j},\ell)\le\ell,\qquad\forall0\le\ell\le m. i ∈ A 0 ∑ P i ( π i , ℓ ) + j ∈ J 0 ∑ P u ( j ) ( π u ∗ , j , ℓ ) ≤ ℓ , ∀0 ≤ ℓ ≤ m . As a result, applying Lemma 2.2 with K ≤ h ∗ K \le h_* K ≤ h ∗ Π I \Pi_I Π I { 1 , … , h ∗ } \{1,\ldots,h_*\} { 1 , … , h ∗ }
Given Corollary 5.2, our next step is to complete head profiles Π I = { π i } i ∈ I \Pi_I=\{\pi_i\}_{i\in I} Π I = { π i } i ∈ I I = A 0 + U 0 ( J 0 ) I=A_0+U_0(J_0) I = A 0 + U 0 ( J 0 ) Ψ I = { ψ i } i ∈ I \Psi_I=\{\psi_i\}_{i\in I} Ψ I = { ψ i } i ∈ I i ∈ I i\in I i ∈ I v ( π i ) + v ( ψ i ) ≥ 1 − 2 / q v(\pi_i)+v(\psi_i)\ge1-2/q v ( π i ) + v ( ψ i ) ≥ 1 − 2/ q T ∗ = 0 T_*=0 T ∗ = 0 T ∗ > 0 T_*>0 T ∗ > 0
In the large tail case, agent i i i π i \pi_i π i Δ ( π i ) \Delta(\pi_i) Δ ( π i ) τ i : = τ i ( π i ) \tau_i:=\tau_i(\pi_i) τ i := τ i ( π i )
τ i ( π i ) : = { Δ ( π i ) / T i , π ≠ ⊥ and T i > 0 ; 0 , otherwise. \tau_i(\pi_i):=
\begin{cases}
\Delta(\pi_i)/T_i, & \pi\ne\bot\text{ and }T_i>0;\\
0, & \text{otherwise.}
\end{cases} τ i ( π i ) := { Δ ( π i ) / T i , 0 , π = ⊥ and T i > 0 ; otherwise. So, we construct a tail profile by taking the last item in each tail segment of length roughly 1 / τ i 1/\tau_i 1/ τ i 1 / q 1/q 1/ q
We would like to repeat this construction on the common tail { h ∗ + 1 , … , m } \{h_*+1,\ldots,m\} { h ∗ + 1 , … , m } h ∗ h_* h ∗ i ∈ I i\in I i ∈ I 1 / q 1/q 1/ q 1 / q 1/q 1/ q
Note that from the perspective of tail profiles, all items of value more than 1 / q 1/q 1/ q 1 / q 1/q 1/ q 1 / q 1/q 1/ q 1 / q 1/q 1/ q i ∈ I i\in I i ∈ I e ∈ M e\in M e ∈ M
v ˉ i ( e ) : = min { v i ( e ) , 1 / q } . \bar{v}_i(e):=\min\{v_i(e),1/q\}. v ˉ i ( e ) := min { v i ( e ) , 1/ q } . and extend it additively. We will construct tail demands based off the item values according to capped valuations v ˉ i \bar{v}_i v ˉ i v i v_i v i
Observe that for every i ∈ I i\in I i ∈ I
v ˉ i ( M ) = ∑ e = 1 m min { v i ( e ) , 1 / q } = h i q + T i . \bar{v}_i(M)=\sum_{e=1}^{m}\min\{v_i(e),1/q\}=\frac{h_i}{q}+T_i. v ˉ i ( M ) = e = 1 ∑ m min { v i ( e ) , 1/ q } = q h i + T i . This is the total capped value of items M M M i i i { h ∗ + 1 , … , m } \{h_*+1,\ldots,m\} { h ∗ + 1 , … , m } h i h_i h i h ∗ h_* h ∗ i i i i i i h i / q + T i h_i/q+T_i h i / q + T i
At the same time, for every agent i ∈ I i\in I i ∈ I h ∗ h_* h ∗ h ∗ / q h_*/q h ∗ / q i i i
∑ e > h ∗ v ˉ i ( e ) = v ˉ i ( M ) − v ˉ i ( { 1 , … , h ∗ } ) ≥ h i q + T i − h ∗ q . \sum_{e>h_*}\bar{v}_i(e)=\bar{v}_i(M)-\bar{v}_i(\{1,\ldots,h_*\})\ge\frac{h_i}{q}+T_i-\frac{h_*}{q}. e > h ∗ ∑ v ˉ i ( e ) = v ˉ i ( M ) − v ˉ i ({ 1 , … , h ∗ }) ≥ q h i + T i − q h ∗ . By Corollary 5.2, the total deficit of agents i ∈ I i\in I i ∈ I ∑ i ∈ I Δ ( π i ) ≤ T ∗ \sum_{i\in I}\Delta(\pi_i)\le T_* ∑ i ∈ I Δ ( π i ) ≤ T ∗ i ∈ I i\in I i ∈ I { h ∗ + 1 , … , m } \{h_*+1,\ldots,m\} { h ∗ + 1 , … , m } T ∗ T_* T ∗
h i q + T i ≥ h ∗ q + T ∗ . \frac{h_i}{q}+T_i\ge\frac{h_*}{q}+T_*. q h i + T i ≥ q h ∗ + T ∗ . This is exactly the condition that defines our choice of supplier u ∗ ∈ U 0 u_*\in U_0 u ∗ ∈ U 0 u ∗ ∈ U 0 u_*\in U_0 u ∗ ∈ U 0
u ∗ = argmin u ∈ U 0 ( h u q + T u ) . u_*=\underset{u\in U_0}{\operatorname{argmin}}\left(\frac{h_u}{q}+T_u\right). u ∗ = u ∈ U 0 argmin ( q h u + T u ) . Since all i ∈ A 0 i \in A_0 i ∈ A 0 Δ ( π i ) = 0 \Delta(\pi_i) = 0 Δ ( π i ) = 0 i ∈ I i \in I i ∈ I U 0 ( J 0 ) ⊆ U 0 U_0(J_0) \subseteq U_0 U 0 ( J 0 ) ⊆ U 0 u ∗ u_* u ∗ i ∈ I i \in I i ∈ I Δ ( π i ) > 0 \Delta(\pi_i) > 0 Δ ( π i ) > 0
v ‾ i ( { h ∗ + 1 , … , m } ) ≥ h i q + T i − h ∗ q ≥ T ∗ . \overline{v}_i(\{h_*+1,\ldots,m\}) \ge\frac{h_i}{q}+T_i-\frac{h_*}{q} \ge T_*. v i ({ h ∗ + 1 , … , m }) ≥ q h i + T i − q h ∗ ≥ T ∗ . It follows that, in order to cover the deficit Δ ( π i ) > 0 \Delta(\pi_i)>0 Δ ( π i ) > 0 i ∈ I i \in I i ∈ I τ i \tau_i τ i τ i , ∗ \tau_{i,*} τ i , ∗
τ i , ∗ : = Δ ( π i ) T ∗ . \tau_{i,*} := \frac{\Delta(\pi_i)}{T_*}. τ i , ∗ := T ∗ Δ ( π i ) . Indeed, the derivations above imply
∑ e > h ∗ τ i , ∗ ⋅ v ‾ i ( e ) ≥ τ i , ∗ ⋅ T ∗ = Δ ( π i ) . \sum_{e>h_*}\tau_{i,*}\cdot\overline{v}_i(e)\ge\tau_{i,*}\cdot T_*=\Delta(\pi_i). e > h ∗ ∑ τ i , ∗ ⋅ v i ( e ) ≥ τ i , ∗ ⋅ T ∗ = Δ ( π i ) . Furthermore, Corollary 5.2 guarantees us that
∑ i ∈ I τ i , ∗ = ∑ i ∈ I Δ ( π i ) T ∗ ≤ 1 , \sum_{i\in I}\tau_{i,*}=\sum_{i\in I}\frac{\Delta(\pi_i)}{T_*}\le1, i ∈ I ∑ τ i , ∗ = i ∈ I ∑ T ∗ Δ ( π i ) ≤ 1 , so all fractional choices of all agents i ∈ I i \in I i ∈ I [5 , 12 ] ), where one would assign every tail item e > h ∗ e>h_* e > h ∗ i ∈ I i\in I i ∈ I τ i , ∗ \tau_{i,*} τ i , ∗ 1 / q 1/q 1/ q
To keep the proof self-contained, instead of relying on faithful rounding we will prove the common tail version of earlier Lemma 3.1, which gives us exactly the same guarantees.
Lemma 5.2 (Lemma 3.1 for a common tail). Let I ⊆ N I\subseteq N I ⊆ N π i \pi_i π i head profile of agent i ∈ I i\in I i ∈ I Π I : = { π i } i ∈ I \Pi_I:=\{\pi_i\}_{i\in I} Π I := { π i } i ∈ I 0 ≤ h ∗ ≤ m 0\le h_*\le m 0 ≤ h ∗ ≤ m T ∗ ≥ 0 T_*\ge0 T ∗ ≥ 0
all demands in Π I \Pi_I Π I { 1 , … , h ∗ } \{1,\ldots,h_*\} { 1 , … , h ∗ }
for every i ∈ I i\in I i ∈ I Δ ( π i ) > 0 \Delta(\pi_i)>0 Δ ( π i ) > 0 h i / q + T i ≥ h ∗ / q + T ∗ h_i/q+T_i\ge h_*/q+T_* h i / q + T i ≥ h ∗ / q + T ∗
∑ i ∈ I Δ ( π i ) ≤ T ∗ \sum_{i\in I}\Delta(\pi_i)\le T_* ∑ i ∈ I Δ ( π i ) ≤ T ∗
there exists a collection Ψ I : = { ψ i } i ∈ I \Psi_I:=\{\psi_i\}_{i\in I} Ψ I := { ψ i } i ∈ I i i i ψ i \psi_i ψ i
∀ 0 ≤ ℓ ≤ m : P ( Π I , ℓ ) + P ( Ψ I , ℓ ) ≤ ℓ \forall0\le\ell\le m:\quad\mathcal{P}(\Pi_I,\ell)+\mathcal{P}(\Psi_I,\ell)\le\ell ∀0 ≤ ℓ ≤ m : P ( Π I , ℓ ) + P ( Ψ I , ℓ ) ≤ ℓ and v ( π i ) + v ( ψ i ) ≥ 1 − 2 / q v(\pi_i)+v(\psi_i)\ge1-2/q v ( π i ) + v ( ψ i ) ≥ 1 − 2/ q i ∈ I i\in I i ∈ I
Proof. If T ∗ = 0 T_*=0 T ∗ = 0 ψ i = ∅ \psi_i=\varnothing ψ i = ∅ i ∈ I i\in I i ∈ I T ∗ > 0 T_*>0 T ∗ > 0 i ∈ I i\in I i ∈ I τ i , ∗ : = Δ ( π i ) / T ∗ \tau_{i,*}:=\Delta(\pi_i)/T_* τ i , ∗ := Δ ( π i ) / T ∗ τ i , ∗ = 0 \tau_{i,*}=0 τ i , ∗ = 0 ψ i = ∅ \psi_i=\varnothing ψ i = ∅ τ i , ∗ > 0 \tau_{i,*}>0 τ i , ∗ > 0
Note that T ∗ > 0 T_*>0 T ∗ > 0 h ∗ < m h_*<m h ∗ < m ψ i \psi_i ψ i τ i , ∗ \tau_{i,*} τ i , ∗ { h ∗ + 1 , … , m } \{h_*+1,\ldots,m\} { h ∗ + 1 , … , m } 1 / τ i , ∗ 1/\tau_{i,*} 1/ τ i , ∗ capped value v ‾ i \overline{v}_i v i ψ i \psi_i ψ i t = 1 , … , ⌊ τ i , ∗ ⋅ ( m − h ∗ ) ⌋ t=1,\ldots,\lfloor\tau_{i,*}\cdot(m-h_*)\rfloor t = 1 , … , ⌊ τ i , ∗ ⋅ ( m − h ∗ )⌋ v ‾ i ( e i , t ) \overline{v}_i(e_{i,t}) v i ( e i , t ) ψ i \psi_i ψ i
e i , t = h ∗ + ⌈ t τ i , ∗ ⌉ . e_{i,t}=h_*+\left\lceil\frac{t}{\tau_{i,*}}\right\rceil. e i , t = h ∗ + ⌈ τ i , ∗ t ⌉ . As in Lemma 3.1, we can represent this construction equivalently as follows. For 1 ≤ k ≤ m − h ∗ 1\le k\le m-h_* 1 ≤ k ≤ m − h ∗ β i , k : = v ‾ i ( h ∗ + k ) \beta_{i,k}:=\overline{v}_i(h_*+k) β i , k := v i ( h ∗ + k ) k ∈ [ m − h ∗ ] k\in[m-h_*] k ∈ [ m − h ∗ ]
⌊ τ i , ∗ ⋅ k ⌋ − ⌊ τ i , ∗ ⋅ ( k − 1 ) ⌋ \lfloor\tau_{i,*}\cdot k\rfloor-\lfloor\tau_{i,*}\cdot(k-1)\rfloor ⌊ τ i , ∗ ⋅ k ⌋ − ⌊ τ i , ∗ ⋅ ( k − 1 )⌋ copies of β i , k \beta_{i,k} β i , k ψ i \psi_i ψ i k k k ⌊ τ i , ∗ ⋅ k ⌋ \lfloor\tau_{i,*} \cdot k \rfloor ⌊ τ i , ∗ ⋅ k ⌋ ψ i \psi_i ψ i
v ( ψ i ) = ∑ k = 1 m − h ∗ β i , k ⋅ ( ⌊ τ i , ∗ ⋅ k ⌋ − ⌊ τ i , ∗ ⋅ ( k − 1 ) ⌋ ) = ∑ k = 1 m − h ∗ ⌊ τ i , ∗ ⋅ k ⌋ ⋅ ( β i , k − β i , k + 1 ) ≥ ∑ k = 1 m − h ∗ ( τ i , ∗ k − 1 ) ⋅ ( β i , k − β i , k + 1 ) = τ i , ∗ ∑ k = 1 m − h ∗ β i , k − β i , 1 . \begin{aligned}
v(\psi_i) = \sum_{k=1}^{m-h_*} \beta_{i,k} \cdot\left(\lfloor\tau_{i,*} \cdot k \rfloor- \lfloor\tau_{i,*} \cdot(k-1) \rfloor\right) = \sum_{k=1}^{m-h_*} \lfloor\tau_{i,*} \cdot k \rfloor\cdot\left(\beta_{i,k} - \beta_{i,k+1}\right) \\
&\ge\sum_{k=1}^{m-h_*} (\tau_{i,*} k - 1) \cdot\left(\beta_{i,k} - \beta_{i,k+1}\right) = \tau_{i,*} \sum_{k=1}^{m-h_*} \beta_{i,k} - \beta_{i,1}.
\end{aligned} v ( ψ i ) = k = 1 ∑ m − h ∗ β i , k ⋅ ( ⌊ τ i , ∗ ⋅ k ⌋ − ⌊ τ i , ∗ ⋅ ( k − 1 )⌋ ) = k = 1 ∑ m − h ∗ ⌊ τ i , ∗ ⋅ k ⌋ ⋅ ( β i , k − β i , k + 1 ) ≥ k = 1 ∑ m − h ∗ ( τ i , ∗ k − 1 ) ⋅ ( β i , k − β i , k + 1 ) = τ i , ∗ k = 1 ∑ m − h ∗ β i , k − β i , 1 . As we derived earlier, under the condition h i / q + T i ≥ h ∗ / q + T ∗ h_i/q + T_i \ge h_*/q + T_* h i / q + T i ≥ h ∗ / q + T ∗
τ i , ∗ ∑ k = 1 m − h ∗ β i , k = τ i , ∗ ⋅ v ˉ i ( { h ∗ + 1 , … , m } ) ≥ τ i , ∗ ⋅ T ∗ = Δ ( π i ) , \tau_{i,*} \sum_{k=1}^{m-h_*} \beta_{i,k} = \tau_{i,*} \cdot\bar{v}_i(\{h_*+1,\ldots,m\}) \ge\tau_{i,*} \cdot T_* = \Delta(\pi_i), τ i , ∗ k = 1 ∑ m − h ∗ β i , k = τ i , ∗ ⋅ v ˉ i ({ h ∗ + 1 , … , m }) ≥ τ i , ∗ ⋅ T ∗ = Δ ( π i ) , therefore, using β i , 1 = v ˉ i ( h ∗ + 1 ) ≤ 1 / q \beta_{i,1} = \bar{v}_i(h_*+1) \le1/q β i , 1 = v ˉ i ( h ∗ + 1 ) ≤ 1/ q
v ( ψ i ) ≥ Δ ( π i ) − β i , 1 ≥ Δ ( π i ) − 1 / q . v(\psi_i) \ge\Delta(\pi_i) - \beta_{i,1} \ge\Delta(\pi_i) - 1/q. v ( ψ i ) ≥ Δ ( π i ) − β i , 1 ≥ Δ ( π i ) − 1/ q . Since Δ ( π i ) = ( 1 − 1 / q − v ( π i ) ) + \Delta(\pi_i) = (1 - 1/q - v(\pi_i))_+ Δ ( π i ) = ( 1 − 1/ q − v ( π i ) ) + v ( π i ) + v ( ψ i ) ≥ 1 − 2 / q v(\pi_i) + v(\psi_i) \ge1 - 2/q v ( π i ) + v ( ψ i ) ≥ 1 − 2/ q
It remains to bound the additional prefix load generated by Ψ I = { ψ i } i ∈ I \Psi_I = \{\psi_i\}_{i\in I} Ψ I = { ψ i } i ∈ I 0 ≤ ℓ ≤ h ∗ 0 \le\ell\le h_* 0 ≤ ℓ ≤ h ∗ P i ( ψ i , ℓ ) = 0 \mathcal{P}_i(\psi_i,\ell) = 0 P i ( ψ i , ℓ ) = 0 i ∈ I i \in I i ∈ I ℓ > h ∗ \ell> h_* ℓ > h ∗ ψ i \psi_i ψ i k = ℓ − h ∗ k = \ell- h_* k = ℓ − h ∗ ⌊ τ i , ∗ ⋅ k ⌋ ≤ τ i , ∗ ⋅ k \lfloor\tau_{i,*} \cdot k \rfloor\le\tau_{i,*} \cdot k ⌊ τ i , ∗ ⋅ k ⌋ ≤ τ i , ∗ ⋅ k β i , k = v ˉ i ( h ∗ + k ) ≤ v i ( h ∗ + k ) \beta_{i,k} = \bar{v}_i(h_*+k) \le v_i(h_*+k) β i , k = v ˉ i ( h ∗ + k ) ≤ v i ( h ∗ + k ) m i ( β i , k ) ≥ h ∗ + k m_i(\beta_{i,k}) \ge h_*+k m i ( β i , k ) ≥ h ∗ + k ℓ \ell ℓ ℓ \ell ℓ Ψ I \Psi_I Ψ I ℓ \ell ℓ
P ( Ψ I , ℓ ) = ∑ i ∈ I P i ( ψ i , ℓ ) ≤ ∑ i ∈ I τ i , ∗ ⋅ ( ℓ − h ∗ ) + = ( ℓ − h ∗ ) + ∑ i ∈ I Δ ( π i ) T ∗ ≤ ( ℓ − h ∗ ) + . \mathcal{P}(\Psi_I,\ell) = \sum_{i\in I} \mathcal{P}_i(\psi_i,\ell) \le\sum_{i\in I} \tau_{i,*} \cdot(\ell-h_*)_+ = (\ell-h_*)_+ \sum_{i\in I} \frac{\Delta(\pi_i)}{T_*} \le(\ell-h_*)_+. P ( Ψ I , ℓ ) = i ∈ I ∑ P i ( ψ i , ℓ ) ≤ i ∈ I ∑ τ i , ∗ ⋅ ( ℓ − h ∗ ) + = ( ℓ − h ∗ ) + i ∈ I ∑ T ∗ Δ ( π i ) ≤ ( ℓ − h ∗ ) + . Since all demands in Π I \Pi_I Π I { 1 , … , h ∗ } \{1,\ldots,h_*\} { 1 , … , h ∗ }
P ( Π I , ℓ ) ≤ min { ℓ , h ∗ } . \mathcal{P}(\Pi_I,\ell) \le\min\{\ell,h_*\}. P ( Π I , ℓ ) ≤ min { ℓ , h ∗ } . for every 0 ≤ ℓ ≤ m 0 \le\ell\le m 0 ≤ ℓ ≤ m
P ( Π I , ℓ ) + P ( Ψ I , ℓ ) ≤ min { ℓ , h ∗ } + ( ℓ − h ∗ ) + = ℓ \mathcal{P}(\Pi_I,\ell) + \mathcal{P}(\Psi_I,\ell) \le\min\{\ell,h_*\} + (\ell-h_*)_+ = \ell P ( Π I , ℓ ) + P ( Ψ I , ℓ ) ≤ min { ℓ , h ∗ } + ( ℓ − h ∗ ) + = ℓ for every $0 ≤ ℓ ≤ m \le\ell\le m ≤ ℓ ≤ m
Lemma 5.2 gives us the particular choice of supplier u ∗ ∈ U 0 u_* \in U_0 u ∗ ∈ U 0 u ∗ ∈ U 0 u_* \in U_0 u ∗ ∈ U 0 i ∈ I = A 0 ∪ U 0 ( J 0 ) i \in I = A_0 \cup U_0(J_0) i ∈ I = A 0 ∪ U 0 ( J 0 ) Δ ( π i ) > 0 \Delta(\pi_i) > 0 Δ ( π i ) > 0 h i / q + T i ≥ h ∗ / q + T ∗ h_i/q + T_i \ge h_*/q + T_* h i / q + T i ≥ h ∗ / q + T ∗ i ∈ A 0 i \in A_0 i ∈ A 0 Δ ( π i ) = 0 \Delta(\pi_i) = 0 Δ ( π i ) = 0
u ∗ = argmin u ∈ U 0 ( h u q + T u ) . u_* = \underset{u\in U_0}{\operatorname{argmin}}\left(\frac{h_u}{q} + T_u\right). u ∗ = u ∈ U 0 argmin ( q h u + T u ) . All previous conclusions of Theorem 5.1, Corollary 5.1 and Corollary 5.2 still apply. We can now give a full proof of the small-tail case.
Theorem 5.2 (Small tail). For every fixed ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 ) q ≥ 3 q \ge3 q ≥ 3 n ε , q n_{\varepsilon,q} n ε , q n ≥ n ( ε , q ) n \ge n(\varepsilon,q) n ≥ n ( ε , q ) γ : = min ( ε , 1 / ( 2 q ) ) \gamma:= \min(\varepsilon,1/(2q)) γ := min ( ε , 1/ ( 2 q )) λ = 2 q 2 q − 1 \lambda= \frac{2q}{2q-1} λ = 2 q − 1 2 q
ρ q , n : = 8 + 4 log λ ( q 3 d 2 ( d + 2 ) n ) + 8 q log ( ( d + 2 ) n ) . \rho_{q,n} := 8 + 4\log_{\lambda}\left(q^3d^2(d+2)n\right) + 8q\log\left((d+2)n\right). ρ q , n := 8 + 4 log λ ( q 3 d 2 ( d + 2 ) n ) + 8 q log ( ( d + 2 ) n ) . If
T ‾ : = 1 n ∑ i ∈ N T i < ρ q , n γ , \overline{T} := \frac{1}{n}\sum_{i\in N} T_i < \frac{\rho_{q,n}}{\gamma}, T := n 1 i ∈ N ∑ T i < γ ρ q , n , there exists I ⊆ N I\subseteq N I ⊆ N ∣ I ∣ ≥ d |I|\ge d ∣ I ∣ ≥ d Π I = { π i } i ∈ I \Pi_I=\{\pi_i\}_{i\in I} Π I = { π i } i ∈ I Ψ I = { ψ i } i ∈ I \Psi_I=\{\psi_i\}_{i\in I} Ψ I = { ψ i } i ∈ I i ∈ I i\in I i ∈ I ( π i , ψ i ) (\pi_i,\psi_i) ( π i , ψ i ) i ∈ I i\in I i ∈ I
v ( π i ) + v ( ψ i ) ≥ 1 − 2 / q v(\pi_i)+v(\psi_i)\ge1-2/q v ( π i ) + v ( ψ i ) ≥ 1 − 2/ q and for every prefix 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m
P ( Π I , ℓ ) + P ( Ψ I , ℓ ) ≤ ℓ . \mathcal{P}(\Pi_I,\ell)+\mathcal{P}(\Psi_I,\ell)\le\ell. P ( Π I , ℓ ) + P ( Ψ I , ℓ ) ≤ ℓ . Proof. Choose n ε , q n_{\varepsilon,q} n ε , q N = U + V + W N=U+V+W N = U + V + W M = E + S M=E+S M = E + S V V V E E E A ⊆ V A\subseteq V A ⊆ V { π i } i ∈ A \{\pi_i\}_{i\in A} { π i } i ∈ A E E E Δ ( π i ) = 0 \Delta(\pi_i)=0 Δ ( π i ) = 0 i ∈ A i\in A i ∈ A ∣ A ∣ ≥ d |A|\ge d ∣ A ∣ ≥ d I ⊆ A I\subseteq A I ⊆ A d d d ψ i = ∅ \psi_i=\varnothing ψ i = ∅ i ∈ I i\in I i ∈ I
Suppose that ∣ A ∣ < d |A|<d ∣ A ∣ < d U 0 ⊆ U U_0\subseteq U U 0 ⊆ U u ∗ ∈ U 0 u_*\in U_0 u ∗ ∈ U 0
u ∗ = arg min u ∈ U 0 ( h u q + T u ) . u_*=\underset{u\in U_0}{\arg\min}\left(\frac{h_u}{q}+T_u\right). u ∗ = u ∈ U 0 arg min ( q h u + T u ) . For this supplier, Corollary 5.1 provides sets A 0 ⊆ A A_0\subseteq A A 0 ⊆ A J 0 ⊆ [ d ] J_0\subseteq[d] J 0 ⊆ [ d ] u ( j ) u(j) u ( j ) u ∗ u_* u ∗ u ( j ) ∈ U 0 u(j)\in U_0 u ( j ) ∈ U 0 ∣ A 0 ∣ + ∣ J 0 ∣ = d |A_0|+|J_0|=d ∣ A 0 ∣ + ∣ J 0 ∣ = d
∑ i ∈ A 0 P i ( π i , ℓ ) + ∑ j ∈ J 0 P u ( j ) ( π u ∗ , j , ℓ ) ≤ ℓ , ∀ 0 ≤ ℓ ≤ m . \sum_{i\in A_0}\mathcal{P}_i(\pi_i,\ell)+\sum_{j\in J_0}\mathcal{P}_{u(j)}(\pi_{u_*,j},\ell)\le\ell,\qquad\forall0\le\ell\le m. i ∈ A 0 ∑ P i ( π i , ℓ ) + j ∈ J 0 ∑ P u ( j ) ( π u ∗ , j , ℓ ) ≤ ℓ , ∀0 ≤ ℓ ≤ m . Set U 0 ( J 0 ) : = { u ( j ) : j ∈ J 0 } U_0(J_0):=\{u(j):j\in J_0\} U 0 ( J 0 ) := { u ( j ) : j ∈ J 0 } I = A 0 ∪ U 0 ( J 0 ) I=A_0\cup U_0(J_0) I = A 0 ∪ U 0 ( J 0 ) π u ( j ) : = π u ∗ , j \pi_{u(j)}:=\pi_{u_*,j} π u ( j ) := π u ∗ , j j ∈ J 0 j\in J_0 j ∈ J 0 Π I : = { π i } i ∈ I \Pi_I:=\{\pi_i\}_{i\in I} Π I := { π i } i ∈ I Π I \Pi_I Π I { 1 , … , h ∗ } \{1,\ldots,h_*\} { 1 , … , h ∗ } h ∗ = h u ∗ h_*=h_{u_*} h ∗ = h u ∗ T ∗ = T u ∗ T_*=T_{u_*} T ∗ = T u ∗
∑ i ∈ I Δ ( π i ) = ∑ i ∈ A 0 Δ ( π i ) + ∑ j ∈ J 0 Δ ( π u ∗ , j ) ≤ T ∗ . \sum_{i\in I}\Delta(\pi_i)=\sum_{i\in A_0}\Delta(\pi_i)+\sum_{j\in J_0}\Delta(\pi_{u_*,j})\le T_*. i ∈ I ∑ Δ ( π i ) = i ∈ A 0 ∑ Δ ( π i ) + j ∈ J 0 ∑ Δ ( π u ∗ , j ) ≤ T ∗ . Since Δ ( π i ) = 0 \Delta(\pi_i)=0 Δ ( π i ) = 0 i ∈ A 0 i\in A_0 i ∈ A 0 U 0 ( J 0 ) ⊆ U 0 U_0(J_0)\subseteq U_0 U 0 ( J 0 ) ⊆ U 0 u ∗ u_* u ∗
h i q + T i ≥ h u ∗ q + T u ∗ = h ∗ q + T ∗ . \frac{h_i}{q}+T_i\ge\frac{h_{u_*}}{q}+T_{u_*}=\frac{h_*}{q}+T_*. q h i + T i ≥ q h u ∗ + T u ∗ = q h ∗ + T ∗ . So, we can apply Lemma 5.2. We obtain a collection Ψ I = { ψ i } i ∈ I \Psi_I=\{\psi_i\}_{i\in I} Ψ I = { ψ i } i ∈ I
P ( Π I , ℓ ) + P ( Ψ I , ℓ ) ≤ ℓ , ∀ 0 ≤ ℓ ≤ m , \mathcal{P}(\Pi_I,\ell)+\mathcal{P}(\Psi_I,\ell)\le\ell,\qquad\forall0\le\ell\le m, P ( Π I , ℓ ) + P ( Ψ I , ℓ ) ≤ ℓ , ∀0 ≤ ℓ ≤ m , and v ( π i ) + v ( ψ i ) ≥ 1 − 2 / q v(\pi_i)+v(\psi_i)\ge1-2/q v ( π i ) + v ( ψ i ) ≥ 1 − 2/ q i ∈ I i\in I i ∈ I ∣ I ∣ = ∣ A 0 ∣ + ∣ J 0 ∣ = d |I|=|A_0|+|J_0|=d ∣ I ∣ = ∣ A 0 ∣ + ∣ J 0 ∣ = d
Combining Theorem 5.2 with Lemma 2.2 proves Theorem 3.1 in the small tail case.
Concluding thoughts
In the previous sections, we proved our main result Theorem 1.1, which we restate below.
Theorem 6.1 (Theorem 1.1 restated). For every fixed ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 ) and integer q ≥ 3 q \ge3 q ≥ 3 there exists n ( ε , q ) n(\varepsilon,q) n ( ε , q ) such that for every n ≥ n ( ε , q ) n \ge n(\varepsilon,q) n ≥ n ( ε , q ) the following holds.
Consider a fair allocation instance with m m m M M M n n n N N N i ∈ N i \in N i ∈ N v i v_i v i i ∈ N i \in N i ∈ N MMS ( M , v i , d ) = 1 \operatorname{MMS}(M,v_i,d)=1 MMS ( M , v i , d ) = 1 1 ≤ d ≤ ( 1 − ε ) n 1 \le d \le(1-\varepsilon)n 1 ≤ d ≤ ( 1 − ε ) n I = { i 1 , … , i d } ⊆ N I=\{i_1,\ldots,i_d\}\subseteq N I = { i 1 , … , i d } ⊆ N ∣ I ∣ = d |I|=d ∣ I ∣ = d ( B i 1 , … , B i d ) (B_{i_1},\ldots,B_{i_d}) ( B i 1 , … , B i d ) M M M I I I i ∈ I i \in I i ∈ I v i ( B i ) ≥ 1 − 2 / q v_i(B_i)\ge1-2/q v i ( B i ) ≥ 1 − 2/ q
An immediate corollary of Theorem 6.1 is that there always exist ( 1 − δ ) (1-\delta) ( 1 − δ ) m m m ( 2 − ε ) n (2-\varepsilon)n ( 2 − ε ) n n ≥ n ε , δ n \ge n_{\varepsilon,\delta} n ≥ n ε , δ
Corollary 6.1 (Corollary 1.1 restated). For every fixed ε , δ ∈ ( 0 , 1 ) \varepsilon,\delta\in(0,1) ε , δ ∈ ( 0 , 1 ) there exists n ε , δ n_{\varepsilon,\delta} n ε , δ such that for every n ≥ n ε , δ n \ge n_{\varepsilon,\delta} n ≥ n ε , δ the following holds. Consider a fair allocation instance with m m m M M M n n n N N N i ∈ N i \in N i ∈ N v i v_i v i m ≤ ( 2 − ε ) n m \le(2-\varepsilon)n m ≤ ( 2 − ε ) n ( A 1 , … , A n ) (A_1,\ldots,A_n) ( A 1 , … , A n ) M M M N N N i ∈ N i \in N i ∈ N
v i ( A i ) ≥ ( 1 − δ ) ⋅ MMS ( M , v i , n ) . v_i(A_i) \ge(1-\delta)\cdot\operatorname{MMS}(M,v_i,n). v i ( A i ) ≥ ( 1 − δ ) ⋅ MMS ( M , v i , n ) . Proof. Using standard reductions, without loss of generality we can assume that
for every agent i ∈ N i \in N i ∈ N e ∈ M e \in M e ∈ M v i ( e ) ∈ [ 0 , 1 ] v_i(e) \in[0,1] v i ( e ) ∈ [ 0 , 1 ]
for every i ∈ N i \in N i ∈ N n n n B i : = ( B i , 1 , … , B i , n ) \mathcal{B}_i := (B_{i,1},\ldots,B_{i,n}) B i := ( B i , 1 , … , B i , n ) M M M MMS ( M , v i , n ) \operatorname{MMS}(M,v_i,n) MMS ( M , v i , n )
∀ j ∈ [ n ] : v i ( B i , j ) = 1 ; \forall j \in[n]:\quad v_i(B_{i,j})=1; ∀ j ∈ [ n ] : v i ( B i , j ) = 1 ; items M = { e 1 , … , e m } M=\{e_1,\ldots,e_m\} M = { e 1 , … , e m } i ∈ N i \in N i ∈ N
v i ( e 1 ) ≥ v i ( e 2 ) ≥ ⋯ ≥ v i ( e m ) . v_i(e_1)\ge v_i(e_2)\ge\cdots\ge v_i(e_m). v i ( e 1 ) ≥ v i ( e 2 ) ≥ ⋯ ≥ v i ( e m ) . For every i ∈ N i \in N i ∈ N B i = ( B i , 1 , … , B i , n ) \mathcal{B}_i=(B_{i,1},\ldots,B_{i,n}) B i = ( B i , 1 , … , B i , n ) s i s_i s i B i \mathcal{B}_i B i i ∈ N i \in N i ∈ N
s i + 2 ( n − s i ) ≤ m ⟹ s i ≥ 2 n − m ≥ ε n . s_i+2(n-s_i)\le m \qquad\Longrightarrow\qquad s_i\ge2n-m\ge\varepsilon n. s i + 2 ( n − s i ) ≤ m ⟹ s i ≥ 2 n − m ≥ ε n . Each item of value 1 produces a singleton bundle in agent i i i s i s_i s i M M M i i i s ∗ : = min i ∈ [ n ] s i s_*:=\min_{i\in[n]}s_i s ∗ := min i ∈ [ n ] s i { 1 , … , s ∗ } \{1,\ldots,s_*\} { 1 , … , s ∗ } i ∈ N i \in N i ∈ N s ∗ = n s_*=n s ∗ = n s ∗ s_* s ∗ M M M d : = n − s ∗ d:=n-s_* d := n − s ∗ i ∈ N i \in N i ∈ N B i \mathcal{B}_i B i [ s ∗ ] [s_*] [ s ∗ ] d d d { s ∗ + 1 , … , m } \{s_*+1,\ldots,m\} { s ∗ + 1 , … , m } d ≤ ( 1 − ε ) n d\le(1-\varepsilon)n d ≤ ( 1 − ε ) n N N N { s ∗ + 1 , … , m } \{s_*+1,\ldots,m\} { s ∗ + 1 , … , m } ε \varepsilon ε q : = ⌈ 2 / δ ⌉ q:=\lceil2/\delta\rceil q := ⌈ 2/ δ ⌉ d d d 1 − 2 / q ≥ 1 − δ 1-2/q\ge1-\delta 1 − 2/ q ≥ 1 − δ n − d = s ∗ n-d=s_* n − d = s ∗ [ s ∗ ] [s_*] [ s ∗ ]
On a separate topic, in [ABE+26] the authors related the guarantees for the 1-out-of-d d d ( d ) (d) ( d )
Corollary 6.2. Consider a fair allocation instance with items M M M and n n n additive agents n n n with valuations { v i } i ∈ N \{v_i\}_{i\in N} { v i } i ∈ N It suffices to add one copy of each of at most
O ( m log 2 n n ) O\left(\frac{m\log^2 n}{n}\right) O ( n m log 2 n ) items to guarantee that, for the new instance with extra items, there exists an allocation that gives every agent i ∈ N i\in N i ∈ N a bundle of value at least MMS ( M , v i , n ) \operatorname{MMS}(M,v_i,n) MMS ( M , v i , n )
In particular, when m = O ( n ) m=O(n) m = O ( n ) O ( log 2 n ) O(\log^2 n) O ( log 2 n ) [12 ] , who obtained a similar implication from their 1-out-of-d d d O ( m log n n ) O\left(m\sqrt{\frac{\log n}{n}}\right) O ( m n l o g n )
AI Usage Disclosure
The authors used OpenAI’s ChatGPT-6.1 Sol as a research assistant in some parts of the proof. In particular, it suggested using Dickson’s lemma argument to prove the existence of a replacement step of a constant size in the proof of Theorem 5.1 (Theorem C.1), as well as Onn’s lemma (Theorem D.2 ) to simplify the computation of the corresponding constant C t , q C_{t,q} C t , q S S S § 5.2 (Lemma 4.1). All mathematical arguments, statements, proofs and exposition in the paper were written and verified by the authors, who take full responsibility for the content of this work.
Proof of the unbounded head rounding theorem
Theorem A.1 (Theorem 2.2 restated). Let Π \Pi Π ∣ π ∣ ≤ h |\pi| \le h ∣ π ∣ ≤ h h ≥ 1 h \ge1 h ≥ 1 i ∈ N i \in\mathbb{N} i ∈ N ( x i , Π ) (x_i, \Pi) ( x i , Π ) x = { x i } i ∈ N x = \{x_i\}_{i \in\mathbb{N}} x = { x i } i ∈ N
There exists a collection y = { y i } i ∈ N y = \{y_i\}_{i\in\mathbb{N}} y = { y i } i ∈ N ( y i , Π ) (y_i,\Pi) ( y i , Π ) i , π i,\pi i , π y i ( π ) ∈ { 0 , 1 } y_i(\pi) \in\{0,1\} y i ( π ) ∈ { 0 , 1 } y i ( π ) = 1 y_i(\pi) = 1 y i ( π ) = 1 π ≠ ⊥ \pi\ne\perp π = ⊥ x i ( π ) > 0 x_i(\pi) > 0 x i ( π ) > 0
H ( y , Π , ℓ ) ≤ H ( x , Π , ℓ ) , ∀ 0 ≤ ℓ ≤ m , \mathcal{H}(y,\Pi,\ell) \le\mathcal{H}(x,\Pi,\ell), \qquad\forall0 \le\ell\le m, H ( y , Π , ℓ ) ≤ H ( x , Π , ℓ ) , ∀0 ≤ ℓ ≤ m , T ( y , Π ) ≤ T ( x , Π ) , \mathcal{T}(y,\Pi) \le\mathcal{T}(x,\Pi), T ( y , Π ) ≤ T ( x , Π ) , I ( y , Π ) ≤ I ( x , Π ) , \mathcal{I}(y,\Pi) \le\mathcal{I}(x,\Pi), I ( y , Π ) ≤ I ( x , Π ) , I ( y , Π ) ≥ I ( x , Π ) − ρ ( h , X ) , \mathcal{I}(y,\Pi) \ge\mathcal{I}(x,\Pi) - \rho(h,X), I ( y , Π ) ≥ I ( x , Π ) − ρ ( h , X ) , where X = ∣ { ( i , π ) : 0 < x i ( π ) < 1 } ∣ X = |\{(i,\pi): 0 < x_i(\pi) < 1\}| X = ∣ {( i , π ) : 0 < x i ( π ) < 1 } ∣
ρ ( h , X ) : = 8 + 4 log X ⋅ ( 1 + ln h + ln X ) . \rho(h,X) := 8 + 4\log X \cdot(1 + \ln h + \ln X). ρ ( h , X ) := 8 + 4 log X ⋅ ( 1 + ln h + ln X ) . Proof. If X = 0 X = 0 X = 0 x x x y = x y = x y = x X ≥ 2 X \ge2 X ≥ 2 x x x y y y k k k x k x^k x k x k + 1 x^{k+1} x k + 1 x k + 1 x^{k+1} x k + 1 π ≠ ⊥ \pi\ne\perp π = ⊥ i ∈ N i \in\mathbb{N} i ∈ N x i k ( π ) ≥ 0 x_i^k(\pi) \ge0 x i k ( π ) ≥ 0 π ∈ Π \pi\in\Pi π ∈ Π ∑ π ∈ Π x i k ( π ) = 1 \sum_{\pi\in\Pi} x_i^k(\pi) = 1 ∑ π ∈ Π x i k ( π ) = 1 X k : = { ( i , π ) : 0 < x i k ( π ) < 1 } \mathcal{X}_k := \{(i,\pi): 0 < x_i^k(\pi) < 1\} X k := {( i , π ) : 0 < x i k ( π ) < 1 } x k x^k x k k k k X k : = ∣ X k ∣ X_k := |\mathcal{X}_k| X k := ∣ X k ∣ x 1 : = x x^1 := x x 1 := x X k : = X X_k := X X k := X
Suppose that X k > 16 X_k > 16 X k > 16 x k x^k x k 0 ≤ ℓ ≤ m 0 \le\ell\le m 0 ≤ ℓ ≤ m
p k : = 4 ( 1 + ln ( h X k ) ) X k . p_k := \frac{4(1+\ln(hX_k))}{X_k}. p k := X k 4 ( 1 + ln ( h X k )) . We create an intermediate solution x ~ k \widetilde{x}^k x k ( i , π ) ∈ X k (i,\pi) \in\mathcal{X}_k ( i , π ) ∈ X k π ≠ ⊥ \pi\ne\perp π = ⊥ min { x i k ( π ) , p k } \min\{x_i^k(\pi),p_k\} min { x i k ( π ) , p k } x i k ( π ) x_i^k(\pi) x i k ( π ) x i k ( ⊥ ) x_i^k(\perp) x i k ( ⊥ )
∀ ( i , π ) ∈ X k : x ~ i k ( π ) = x i k ( π ) − min { x i k ( π ) , p k } , x ~ i k ( ⊥ ) = x i k ( ⊥ ) + ∑ ( i , π ) ∈ X k π ≠ ⊥ min { x i k ( π ) , p k } . \forall(i,\pi)\in\mathcal{X}_k:\quad\widetilde{x}_i^k(\pi) = x_i^k(\pi) - \min\{x_i^k(\pi),p_k\}, \qquad\widetilde{x}_i^k(\perp) = x_i^k(\perp) + \sum_{\substack{(i,\pi)\in\mathcal{X}_k\\\pi\ne\perp}} \min\{x_i^k(\pi),p_k\}. ∀ ( i , π ) ∈ X k : x i k ( π ) = x i k ( π ) − min { x i k ( π ) , p k } , x i k ( ⊥ ) = x i k ( ⊥ ) + ( i , π ) ∈ X k π = ⊥ ∑ min { x i k ( π ) , p k } . Let X ~ k = { ( i , π ) : 0 < x ~ i k ( π ) < 1 } \widetilde{\mathcal{X}}_k = \{(i,\pi): 0 < \widetilde{x}_i^k(\pi) < 1\} X k = {( i , π ) : 0 < x i k ( π ) < 1 } x ~ k \widetilde{x}^k x k X ~ k = ∅ \widetilde{\mathcal{X}}_k = \varnothing X k = ∅ x ~ k \widetilde{x}^k x k x k + 1 = x ~ k x^{k+1} = \widetilde{x}^k x k + 1 = x k
If X ~ k ≠ ∅ \widetilde{\mathcal{X}}_k \ne\varnothing X k = ∅ ( i , π ) ∈ X ~ k (i,\pi) \in\widetilde{\mathcal{X}}_k ( i , π ) ∈ X k π ≠ ⊥ \pi\ne\perp π = ⊥ x i k ( π ) > p k x_i^k(\pi) > p_k x i k ( π ) > p k p k p_k p k x ~ k \widetilde{x}^k x k ( i , π ) ∈ X ~ k (i,\pi) \in\widetilde{\mathcal{X}}_k ( i , π ) ∈ X k π ≠ ⊥ \pi\ne\perp π = ⊥ i i i α ∈ π \alpha\in\pi α ∈ π M k \mathcal{M}_k M k
M k : = ⊎ ( i , π ) ∈ X ~ k π ≠ ⊥ { m i ( α ) : α ∈ π } . \mathcal{M}_k := \mathop{\uplus}_{\substack{(i,\pi)\in\widetilde{\mathcal{X}}_k\\\pi\ne\perp}} \{m_i(\alpha):\alpha\in\pi\}. M k := ⊎ ( i , π ) ∈ X k π = ⊥ { m i ( α ) : α ∈ π } . Lemma A.1. It holds I ( x ~ k , Π ) ≥ I ( x k , Π ) − p k X k \mathcal{I}(\widetilde{x}^k,\Pi) \ge\mathcal{I}(x^k,\Pi) - p_kX_k I ( x k , Π ) ≥ I ( x k , Π ) − p k X k T ( x ~ k , Π ) ≤ T ( x k , Π ) \mathcal{T}(\widetilde{x}^k,\Pi) \le\mathcal{T}(x^k,\Pi) T ( x k , Π ) ≤ T ( x k , Π )
∀ 0 ≤ ℓ ≤ m : H ( x ~ k , Π , ℓ ) + p k P k ( ℓ ) ≤ H ( x k , Π , ℓ ) , \forall0 \le\ell\le m:\qquad\mathcal{H}(\widetilde{x}^k,\Pi,\ell) + p_k\mathcal{P}_k(\ell) \le\mathcal{H}(x^k,\Pi,\ell), ∀0 ≤ ℓ ≤ m : H ( x k , Π , ℓ ) + p k P k ( ℓ ) ≤ H ( x k , Π , ℓ ) , where for every − 1 ≤ ℓ ≤ m -1 \le\ell\le m − 1 ≤ ℓ ≤ m
P k ( ℓ ) = ∑ ( i , π ) ∈ X ~ k P i ( π , ℓ ) \mathcal{P}_k(\ell) = \sum_{(i,\pi)\in\widetilde{\mathcal{X}}_k} \mathcal{P}_i(\pi,\ell) P k ( ℓ ) = ( i , π ) ∈ X k ∑ P i ( π , ℓ ) is the total number of breakpoints in M k \mathcal{M}_k M k ℓ \ell ℓ P k ( − 1 ) = 0 \mathcal{P}_k(-1) = 0 P k ( − 1 ) = 0
Proof. Since each non-⊥ \perp ⊥ x k x^k x k p k p_k p k I ( x ~ k , Π ) ≥ I ( x k , Π ) − p k X k \mathcal{I}(\widetilde{x}^k,\Pi) \ge\mathcal{I}(x^k,\Pi)-p_kX_k I ( x k , Π ) ≥ I ( x k , Π ) − p k X k ⊥ \perp ⊥ τ ( x ~ k , Π ) ≤ τ ( x k , Π ) \tau(\widetilde{x}^k,\Pi) \le\tau(x^k,\Pi) τ ( x k , Π ) ≤ τ ( x k , Π )
Finally, by construction of x ~ k \widetilde{x}^k x k 0 ≤ ℓ ≤ m 0 \le\ell\le m 0 ≤ ℓ ≤ m
H ( x k , Π , ℓ ) − H ( x ~ k , Π , ℓ ) = ∑ i ∈ N π ≠ ⊥ ( x i k ( π ) − x ~ i k ( π ) ) ⋅ P i ( π , ℓ ) ≥ ∑ ( i , π ) ∈ X ~ k p k ⋅ P i ( π , ℓ ) . \mathcal{H}(x^k,\Pi,\ell)-\mathcal{H}(\widetilde{x}^k,\Pi,\ell)=\sum_{\substack{i\in N\\ \pi\ne\perp}}(x_i^k(\pi)-\widetilde{x}_i^k(\pi))\cdot\mathcal{P}_i(\pi,\ell)\ge\sum_{(i,\pi)\in\widetilde{X}_k}p_k\cdot\mathcal{P}_i(\pi,\ell). H ( x k , Π , ℓ ) − H ( x k , Π , ℓ ) = i ∈ N π = ⊥ ∑ ( x i k ( π ) − x i k ( π )) ⋅ P i ( π , ℓ ) ≥ ( i , π ) ∈ X k ∑ p k ⋅ P i ( π , ℓ ) . which is exactly p k ⋅ P k ( ℓ ) p_k\cdot\mathcal{P}_k(\ell) p k ⋅ P k ( ℓ )
Next, we will create a small set of head checkpoints , i.e a subset of prefixes ℓ \ell ℓ H ( x k + 1 , Π , ℓ ) ≤ H ( x k , Π , ℓ ) \mathcal{H}(x^{k+1},\Pi,\ell)\le\mathcal{H}(x^k,\Pi,\ell) H ( x k + 1 , Π , ℓ ) ≤ H ( x k , Π , ℓ ) M k \mathcal{M}_k M k m i j m_{ij} m ij j j j M k \mathcal{M}_k M k 1 ≤ j ≤ ∣ M k ∣ 1\le j\le|\mathcal{M}_k| 1 ≤ j ≤ ∣ M k ∣ 1 / j 1/j 1/ j m i j m_{ij} m ij t = − 1 t=-1 t = − 1 ℓ ∈ { 0 , … , m } \ell\in\{0,\ldots,m\} ℓ ∈ { 0 , … , m } ℓ > t \ell>t ℓ > t t + 1 t+1 t + 1 ℓ \ell ℓ M k \mathcal{M}_k M k p k p_k p k ℓ > t \ell>t ℓ > t
∑ j = P k ( t ) + 1 P k ( ℓ ) 1 j > p k . \sum_{j=\mathcal{P}_k(t)+1}^{\mathcal{P}_k(\ell)}\frac{1}{j}>p_k. j = P k ( t ) + 1 ∑ P k ( ℓ ) j 1 > p k . We put a checkpoint at ℓ \ell ℓ t t t ℓ \ell ℓ ℓ > t \ell>t ℓ > t L k \mathcal{L}_k L k − 1 -1 − 1
Lemma A.2. The total number of checkpoints at phase k k k ∣ L k ∣ ≤ X k / 4 |\mathcal{L}_k|\le X_k/4 ∣ L k ∣ ≤ X k /4 z z z z i ( π ) = x ~ i k ( π ) z_i(\pi)=\widetilde{x}_i^k(\pi) z i ( π ) = x i k ( π ) ( i , π ) (i,\pi) ( i , π ) x ~ i k ( π ) ∈ { 0 , 1 } \widetilde{x}_i^k(\pi)\in\{0,1\} x i k ( π ) ∈ { 0 , 1 } ℓ k ∈ L k \ell_k\in\mathcal{L}_k ℓ k ∈ L k H ( z , Π , ℓ k ) ≤ H ( x ~ k , Π , ℓ k ) \mathcal{H}(z,\Pi,\ell_k)\le\mathcal{H}(\widetilde{x}^k,\Pi,\ell_k) H ( z , Π , ℓ k ) ≤ H ( x k , Π , ℓ k ) 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m
H ( z , Π , ℓ ) ≤ H ( x ~ k , Π , ℓ ) + p k P k ( ℓ ) . \mathcal{H}(z,\Pi,\ell)\le\mathcal{H}(\widetilde{x}^k,\Pi,\ell)+p_k\mathcal{P}_k(\ell). H ( z , Π , ℓ ) ≤ H ( x k , Π , ℓ ) + p k P k ( ℓ ) . Proof. If M k = ∅ \mathcal{M}_k=\varnothing M k = ∅ L k = ∅ \mathcal{L}_k=\varnothing L k = ∅ ( i , π ) ∈ X ~ k (i,\pi)\in\widetilde{X}_k ( i , π ) ∈ X k ℓ \ell ℓ M k ≠ ∅ \mathcal{M}_k\ne\varnothing M k = ∅ P k ( m ) ≥ 1 \mathcal{P}_k(m)\ge1 P k ( m ) ≥ 1 L k \mathcal{L}_k L k
∣ L k ∣ ⋅ p k ≤ ∑ j = 1 P k ( m ) 1 j ≤ 1 + ln P k ( m ) ⟹ ∣ L k ∣ ≤ 1 + ln P k ( m ) p k . |\mathcal{L}_k|\cdot p_k\le\sum_{j=1}^{\mathcal{P}_k(m)}\frac{1}{j}\le1+\ln\mathcal{P}_k(m)\quad\Longrightarrow\quad|\mathcal{L}_k|\le\frac{1+\ln\mathcal{P}_k(m)}{p_k}. ∣ L k ∣ ⋅ p k ≤ j = 1 ∑ P k ( m ) j 1 ≤ 1 + ln P k ( m ) ⟹ ∣ L k ∣ ≤ p k 1 + ln P k ( m ) . By assumption, every profile π ∈ Π − ⊥ \pi\in\Pi-\perp π ∈ Π − ⊥ ∣ π ∣ ≤ h |\pi|\le h ∣ π ∣ ≤ h ( i , π ) ∈ X ~ k (i,\pi)\in\widetilde{X}_k ( i , π ) ∈ X k π ≠ ⊥ \pi\ne\perp π = ⊥ x k x^k x k
P k ( m ) = ∣ M k ∣ = ∑ ( i , π ) ∈ X ~ k π ≠ ⊥ ∣ π ∣ ≤ ∑ ( i , π ) ∈ X k π ≠ ⊥ ∣ π ∣ ≤ h X k , \mathcal{P}_k(m)=|\mathcal{M}_k|=\sum_{\substack{(i,\pi)\in\widetilde{X}_k\\ \pi\ne\perp}}|\pi|\le\sum_{\substack{(i,\pi)\in X_k\\ \pi\ne\perp}}|\pi|\le hX_k, P k ( m ) = ∣ M k ∣ = ( i , π ) ∈ X k π = ⊥ ∑ ∣ π ∣ ≤ ( i , π ) ∈ X k π = ⊥ ∑ ∣ π ∣ ≤ h X k , and substituting the definition of p k = ( 1 / X k ) ⋅ 4 ( 1 + ln ( h X k ) ) p_k=(1/X_k)\cdot4(1+\ln(hX_k)) p k = ( 1/ X k ) ⋅ 4 ( 1 + ln ( h X k ))
∣ L k ∣ ≤ 1 + ln P k ( m ) p k ≤ X k ( 1 + ln ( h X k ) ) 4 ( 1 + ln ( h X k ) ) = X k 4 . |\mathcal{L}_k|\le\frac{1+\ln\mathcal{P}_k(m)}{p_k}\le\frac{X_k(1+\ln(hX_k))}{4(1+\ln(hX_k))}=\frac{X_k}{4}. ∣ L k ∣ ≤ p k 1 + ln P k ( m ) ≤ 4 ( 1 + ln ( h X k )) X k ( 1 + ln ( h X k )) = 4 X k . Next, for 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m ℓ k ∈ L k \ell_k\in\mathcal{L}_k ℓ k ∈ L k ℓ \ell ℓ ℓ k ≤ ℓ \ell_k\le\ell ℓ k ≤ ℓ ℓ k = − 1 \ell_k=-1 ℓ k = − 1 z z z x ~ k \widetilde{x}^k x k X ~ k \widetilde{X}_k X k
H ( z , Π , ℓ ) − H ( x ~ k , Π , ℓ ) = H ( z , Π , ℓ k ) − H ( x ~ k , Π , ℓ k ) + ∑ ( i , π ) ∈ X ~ k ( z i ( π ) − x ~ i k ( π ) ) ⋅ ( P i ( π , ℓ ) − P i ( π , ℓ k ) ) . \mathcal{H}(z,\Pi,\ell)-\mathcal{H}(\widetilde{x}^k,\Pi,\ell)=\mathcal{H}(z,\Pi,\ell_k)-\mathcal{H}(\widetilde{x}^k,\Pi,\ell_k)
+\sum_{(i,\pi)\in\widetilde{X}_k}(z_i(\pi)-\widetilde{x}_i^k(\pi))\cdot\left(\mathcal{P}_i(\pi,\ell)-\mathcal{P}_i(\pi,\ell_k)\right). H ( z , Π , ℓ ) − H ( x k , Π , ℓ ) = H ( z , Π , ℓ k ) − H ( x k , Π , ℓ k ) + ( i , π ) ∈ X k ∑ ( z i ( π ) − x i k ( π )) ⋅ ( P i ( π , ℓ ) − P i ( π , ℓ k ) ) . By assumption, H ( z , Π , ℓ k ) ≤ H ( x ~ k , Π , ℓ k ) \mathcal{H}(z,\Pi,\ell_k) \le\mathcal{H}(\widetilde{x}^{k},\Pi,\ell_k) H ( z , Π , ℓ k ) ≤ H ( x k , Π , ℓ k ) z i ( π ) z_i(\pi) z i ( π ) x ~ i k ( π ) \widetilde{x}^{k}_i(\pi) x i k ( π ) [ 0 , 1 ] [0,1] [ 0 , 1 ]
H ( z , Π , ℓ ) − H ( x ~ k , Π , ℓ ) ≤ ∑ ( i , π ) ∈ X ~ k ∣ z i ( π ) − x ~ i k ( π ) ∣ ⋅ ( P i ( π , ℓ ) − P i ( π , ℓ k ) ) ≤ ∑ ( i , π ) ∈ X ~ k ( P i ( π , ℓ ) − P i ( π , ℓ k ) ) = P k ( ℓ ) − P k ( ℓ k ) . \begin{aligned}
\mathcal{H}(z,\Pi,\ell)-\mathcal{H}(\widetilde{x}^{k},\Pi,\ell)
\le\sum_{(i,\pi)\in\widetilde{\mathcal{X}}_k}|z_i(\pi)-\widetilde{x}^{k}_i(\pi)|\cdot\left(\mathcal{P}_i(\pi,\ell)-\mathcal{P}_i(\pi,\ell_k)\right) \\
&\le\sum_{(i,\pi)\in\widetilde{\mathcal{X}}_k}\left(\mathcal{P}_i(\pi,\ell)-\mathcal{P}_i(\pi,\ell_k)\right)
=\mathcal{P}_k(\ell)-\mathcal{P}_k(\ell_k).
\end{aligned} H ( z , Π , ℓ ) − H ( x k , Π , ℓ ) ≤ ( i , π ) ∈ X k ∑ ∣ z i ( π ) − x i k ( π ) ∣ ⋅ ( P i ( π , ℓ ) − P i ( π , ℓ k ) ) ≤ ( i , π ) ∈ X k ∑ ( P i ( π , ℓ ) − P i ( π , ℓ k ) ) = P k ( ℓ ) − P k ( ℓ k ) . If ℓ k = ℓ \ell_k=\ell ℓ k = ℓ P k ( ℓ ) − P k ( ℓ k ) = 0 \mathcal{P}_k(\ell)-\mathcal{P}_k(\ell_k)=0 P k ( ℓ ) − P k ( ℓ k ) = 0 L k \mathcal{L}_k L k
∑ j = P k ( ℓ k ) + 1 P k ( ℓ ) 1 j ≤ p k . \sum_{j=\mathcal{P}_k(\ell_k)+1}^{\mathcal{P}_k(\ell)}\frac{1}{j}\le p_k. j = P k ( ℓ k ) + 1 ∑ P k ( ℓ ) j 1 ≤ p k . When P k ( ℓ ) > 0 \mathcal{P}_k(\ell)>0 P k ( ℓ ) > 0 1 / P k ( ℓ ) 1/\mathcal{P}_k(\ell) 1/ P k ( ℓ )
P k ( ℓ ) − P k ( ℓ k ) P k ( ℓ ) ≤ ∑ j = P k ( ℓ k ) + 1 P k ( ℓ ) 1 j ≤ p k ⟹ P k ( ℓ ) − P k ( ℓ k ) ≤ p k P k ( ℓ ) . \frac{\mathcal{P}_k(\ell)-\mathcal{P}_k(\ell_k)}{\mathcal{P}_k(\ell)}
\le\sum_{j=\mathcal{P}_k(\ell_k)+1}^{\mathcal{P}_k(\ell)}\frac{1}{j}
\le p_k
\quad\Longrightarrow\quad
\mathcal{P}_k(\ell)-\mathcal{P}_k(\ell_k)\le p_k\mathcal{P}_k(\ell). P k ( ℓ ) P k ( ℓ ) − P k ( ℓ k ) ≤ j = P k ( ℓ k ) + 1 ∑ P k ( ℓ ) j 1 ≤ p k ⟹ P k ( ℓ ) − P k ( ℓ k ) ≤ p k P k ( ℓ ) . We conclude that H ( z , Π , ℓ ) − H ( x ~ k , Π , ℓ ) ≤ p k P k ( ℓ ) \mathcal{H}(z,\Pi,\ell)-\mathcal{H}(\widetilde{x}^{k},\Pi,\ell)\le p_k\mathcal{P}_k(\ell) H ( z , Π , ℓ ) − H ( x k , Π , ℓ ) ≤ p k P k ( ℓ )
It follows from Lemma A.1 and Lemma A.2 that, to preserve the prefix load constraints at every prefix 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m k k k L k \mathcal{L}_k L k x ~ k \widetilde{x}^{k} x k z ∗ z^* z ∗
∑ π ∈ Π z i ( π ) = 1 , ∀ i ∈ N , z i ( π ) = x ~ i k ( π ) , ∀ ( i , π ) : x ~ i k ( π ) ∈ { 0 , 1 } , τ ( z , Π ) ≤ τ ( x ~ k , Π ) , H ( z , Π , ℓ k ) ≤ H ( x ~ k , Π , ℓ k ) , ∀ ℓ k ∈ L k , 0 ≤ z i ( π ) ≤ 1 , ∀ i ∈ N , π ∈ Π . \begin{aligned}
\sum_{\pi\in\Pi}z_i(\pi)&=1, &&\forall i\in N,\\
z_i(\pi)&=\widetilde{x}^{k}_i(\pi), &&\forall(i,\pi):\widetilde{x}^{k}_i(\pi)\in\{0,1\},\\
\tau(z,\Pi)&\le\tau(\widetilde{x}^{k},\Pi),\\
\mathcal{H}(z,\Pi,\ell_k)&\le\mathcal{H}(\widetilde{x}^{k},\Pi,\ell_k), &&\forall\ell_k\in\mathcal{L}_k,\\
0&\le z_i(\pi)\le1, &&\forall i\in N,\pi\in\Pi.
\end{aligned} π ∈ Π ∑ z i ( π ) z i ( π ) τ ( z , Π ) H ( z , Π , ℓ k ) 0 = 1 , = x i k ( π ) , ≤ τ ( x k , Π ) , ≤ H ( x k , Π , ℓ k ) , ≤ z i ( π ) ≤ 1 , ∀ i ∈ N , ∀ ( i , π ) : x i k ( π ) ∈ { 0 , 1 } , ∀ ℓ k ∈ L k , ∀ i ∈ N , π ∈ Π. Clearly, z = x ~ k z=\widetilde{x}^{k} z = x k z ∗ z^* z ∗
I ( z ∗ , Π ) = ∑ i ∈ N ∑ π ≠ ⊥ z i ∗ ( π ) . \mathcal{I}(z^*,\Pi)=\sum_{i\in N}\sum_{\pi\ne\bot}z_i^*(\pi). I ( z ∗ , Π ) = i ∈ N ∑ π = ⊥ ∑ z i ∗ ( π ) . Such an extreme point must exist, because (P) is nonempty and compact, i.e bounded and closed. Then, by maximality I ( z ∗ , Π ) ≥ I ( x ~ k , Π ) \mathcal{I}(z^*,\Pi)\ge\mathcal{I}(\widetilde{x}^{k},\Pi) I ( z ∗ , Π ) ≥ I ( x k , Π )
Lemma A.3. Let z ∗ z^* z ∗ z ∗ z^* z ∗
Proof. Let Z ∗ : = ∣ { ( i , π ) : 0 < z i ∗ ( π ) < 1 } ∣ Z^*:=|\{(i,\pi):0<z_i^*(\pi)<1\}| Z ∗ := ∣ {( i , π ) : 0 < z i ∗ ( π ) < 1 } ∣ z ∗ z^* z ∗ i ∈ N i\in N i ∈ N 0 < z i ∗ ( π ′ ) < 1 0<z_i^*(\pi')<1 0 < z i ∗ ( π ′ ) < 1 π ′ ∈ Π \pi'\in\Pi π ′ ∈ Π π ′ ′ ∈ Π \pi''\in\Pi π ′′ ∈ Π 0 < z i ∗ ( π ′ ′ ) < 1 0<z_i^*(\pi'')<1 0 < z i ∗ ( π ′′ ) < 1 ∑ π ∈ Π z i ∗ ( π ) = 1 \sum_{\pi\in\Pi}z_i^*(\pi)=1 ∑ π ∈ Π z i ∗ ( π ) = 1 Z ∗ / 2 Z^*/2 Z ∗ /2 i ∈ N i\in N i ∈ N 0 < z i ∗ ( π ) < 1 0<z_i^*(\pi)<1 0 < z i ∗ ( π ) < 1 π ∈ Π \pi\in\Pi π ∈ Π z i ∗ ( π ) ∈ { 0 , 1 } z_i^*(\pi)\in\{0,1\} z i ∗ ( π ) ∈ { 0 , 1 } ( i , π ) (i,\pi) ( i , π )
Suppose that Z ∗ > X k / 2 + 2 Z^*>X_k/2+2 Z ∗ > X k /2 + 2 ( i , π ) (i,\pi) ( i , π ) 0 < z i ∗ ( π ) < 1 0<z_i^*(\pi)<1 0 < z i ∗ ( π ) < 1 Z ∗ Z^* Z ∗ Z ∗ / 2 Z^*/2 Z ∗ /2 ∑ π ∈ Π z i ∗ ( π ) = 1 \sum_{\pi\in\Pi}z_i^*(\pi)=1 ∑ π ∈ Π z i ∗ ( π ) = 1 i ∈ N i\in N i ∈ N τ ( z ∗ , Π ) \tau(z^*,\Pi) τ ( z ∗ , Π ) H ( z ∗ , Π , ℓ k ) \mathcal{H}(z^*,\Pi,\ell_k) H ( z ∗ , Π , ℓ k ) ℓ k ∈ L k \ell_k \in L_k ℓ k ∈ L k ∣ L k ∣ ≤ X k / 4 |L_k| \le X_k/4 ∣ L k ∣ ≤ X k /4 Z ∗ > X k / 2 + 2 Z^* > X_k/2+2 Z ∗ > X k /2 + 2
Z ∗ 2 + 1 + ∣ L k ∣ ≤ Z ∗ 2 + 4 + X k 4 < Z ∗ 2 + 2 Z ∗ 4 = Z ∗ . \frac{Z^*}{2}+1+|L_k|\le\frac{Z^*}{2}+\frac{4+X_k}{4}<\frac{Z^*}{2}+\frac{2Z^*}{4}=Z^*. 2 Z ∗ + 1 + ∣ L k ∣ ≤ 2 Z ∗ + 4 4 + X k < 2 Z ∗ + 4 2 Z ∗ = Z ∗ . Therefore, the restricted matrix has rank less than Z ∗ Z^* Z ∗ z ∗ z^* z ∗ z ∗ z^* z ∗ z ∗ z^* z ∗
Take this extreme point z ∗ z^* z ∗ x k + 1 = z ∗ x^{k+1}=z^* x k + 1 = z ∗
∀ 0 ≤ ℓ ≤ m : H ( x k + 1 , Π , ℓ ) ≤ H ( x k , Π , ℓ ) , and τ ( x k + 1 , Π ) ≤ τ ( x k , Π ) , \forall0\le\ell\le m:\ \mathcal{H}(x^{k+1},\Pi,\ell)\le\mathcal{H}(x^k,\Pi,\ell),\qquad\text{and}\qquad\tau(x^{k+1},\Pi)\le\tau(x^k,\Pi), ∀0 ≤ ℓ ≤ m : H ( x k + 1 , Π , ℓ ) ≤ H ( x k , Π , ℓ ) , and τ ( x k + 1 , Π ) ≤ τ ( x k , Π ) , as well as
I ( x k , Π ) − I ( x k + 1 , Π ) ≤ p k X k = 4 ( 1 + ln ( h X k ) ) . \mathcal{I}(x^k,\Pi)-\mathcal{I}(x^{k+1},\Pi)\le p_kX_k=4(1+\ln(hX_k)). I ( x k , Π ) − I ( x k + 1 , Π ) ≤ p k X k = 4 ( 1 + ln ( h X k )) . By Lemma A.3 , X k + 1 ≤ X k / 2 + 2 X_{k+1}\le X_k/2+2 X k + 1 ≤ X k /2 + 2 k k k k ≥ 1 k\ge1 k ≥ 1
X k ≤ 4 + ( X − 4 ) ⋅ 2 − ( k − 1 ) . X_k\le4+(X-4)\cdot2^{-(k-1)}. X k ≤ 4 + ( X − 4 ) ⋅ 2 − ( k − 1 ) . Then, if at phase K + 1 K+1 K + 1 X K + 1 ≤ 16 X_{K+1}\le16 X K + 1 ≤ 16 K ≤ log X K\le\log X K ≤ log X x K + 1 x^{K+1} x K + 1 y y y i ∈ N i\in N i ∈ N 0 < x i K + 1 ( π ) < 1 0<x_i^{K+1}(\pi)<1 0 < x i K + 1 ( π ) < 1 π ∈ Π \pi\in\Pi π ∈ Π y i ( ⊥ ) = 1 y_i(\perp)=1 y i ( ⊥ ) = 1 i ∈ N i\in N i ∈ N y i ( π ) = x i K + 1 ( π ) y_i(\pi)=x_i^{K+1}(\pi) y i ( π ) = x i K + 1 ( π ) π ∈ Π \pi\in\Pi π ∈ Π y y y
I ( y , Π ) ≥ I ( x K + 1 , Π ) − 8. \mathcal{I}(y,\Pi)\ge\mathcal{I}(x^{K+1},\Pi)-8. I ( y , Π ) ≥ I ( x K + 1 , Π ) − 8. Summing up the loss in I ( x k , Π ) \mathcal{I}(x^k,\Pi) I ( x k , Π ) k ∈ [ K ] k\in[K] k ∈ [ K ] X k ≤ X X_k\le X X k ≤ X
I ( x , Π ) − I ( y , Π ) ≤ 8 + ∑ k = 1 K ( I ( x k , Π ) − I ( x k + 1 , Π ) ) ≤ 8 + 4 ∑ k = 1 K ( 1 + ln ( h X k ) ) \mathcal{I}(x,\Pi)-\mathcal{I}(y,\Pi)\le8+\sum_{k=1}^{K}\left(\mathcal{I}(x^k,\Pi)-\mathcal{I}(x^{k+1},\Pi)\right)\le8+4\sum_{k=1}^{K}(1+\ln(hX_k)) I ( x , Π ) − I ( y , Π ) ≤ 8 + k = 1 ∑ K ( I ( x k , Π ) − I ( x k + 1 , Π ) ) ≤ 8 + 4 k = 1 ∑ K ( 1 + ln ( h X k )) ≤ 8 + 4 log X ⋅ ( 1 + ln ( h X ) ) = ρ ( h , X ) . \le8+4\log X\cdot(1+\ln(hX))=\rho(h,X). ≤ 8 + 4 log X ⋅ ( 1 + ln ( h X )) = ρ ( h , X ) . To ensure that I ( y , Π ) ≤ I ( x , Π ) \mathcal{I}(y,\Pi)\le\mathcal{I}(x,\Pi) I ( y , Π ) ≤ I ( x , Π ) I ( y , Π ) > I ( x , Π ) \mathcal{I}(y,\Pi)>\mathcal{I}(x,\Pi) I ( y , Π ) > I ( x , Π ) ⌊ I ( x , Π ) ⌋ \lfloor\mathcal{I}(x,\Pi)\rfloor ⌊ I ( x , Π )⌋ y y y ⊥ \perp ⊥ I ( x , Π ) − 1 \mathcal{I}(x,\Pi)-1 I ( x , Π ) − 1 i ∈ N i\in N i ∈ N y i ( π ) = 1 y_i(\pi)=1 y i ( π ) = 1 π ≠ ⊥ \pi\ne\perp π = ⊥ x i ( π ) > 0 x_i(\pi)>0 x i ( π ) > 0
Proof of the bounded head and tail rounding theorem
Theorem B.1 (Theorem 3.2 restated). Let Π \Pi Π ∣ π ∣ ≤ q |\pi|\le q ∣ π ∣ ≤ q q ≥ 3 q\ge3 q ≥ 3 i ∈ N i\in N i ∈ N ( x i , Π ) (x_i,\Pi) ( x i , Π ) x = { x i } i ∈ N x=\{x_i\}_{i\in N} x = { x i } i ∈ N π ∈ Π − ⊥ \pi\in\Pi-\perp π ∈ Π − ⊥ Δ ( π ) = 0 \Delta(\pi)=0 Δ ( π ) = 0 Δ ( π ) ≥ δ > 0 \Delta(\pi)\ge\delta>0 Δ ( π ) ≥ δ > 0 δ ∈ ( 0 , 1 ] \delta\in(0,1] δ ∈ ( 0 , 1 ] λ > 1 \lambda>1 λ > 1 y = { y i } i ∈ N y=\{y_i\}_{i\in N} y = { y i } i ∈ N ( y i , Π ) (y_i,\Pi) ( y i , Π ) i , π i,\pi i , π y i ( π ) ∈ { 0 , 1 } y_i(\pi)\in\{0,1\} y i ( π ) ∈ { 0 , 1 } y i ( π ) = 1 y_i(\pi)=1 y i ( π ) = 1 π ≠ ⊥ \pi\ne\perp π = ⊥ x i ( π ) > 0 x_i(\pi)>0 x i ( π ) > 0
H ( y , Π , ℓ ) ≤ H ( x , Π , ℓ ) , ∀ 0 ≤ ℓ ≤ m , \mathcal{H}(y,\Pi,\ell)\le\mathcal{H}(x,\Pi,\ell),\qquad\forall\,0\le\ell\le m, H ( y , Π , ℓ ) ≤ H ( x , Π , ℓ ) , ∀ 0 ≤ ℓ ≤ m , T ( y , Π , ℓ ) ≤ λ T ( x , Π , ℓ ) , ∀ 0 ≤ ℓ ≤ m , \mathcal{T}(y,\Pi,\ell)\le\lambda\mathcal{T}(x,\Pi,\ell),\qquad\forall\,0\le\ell\le m, T ( y , Π , ℓ ) ≤ λ T ( x , Π , ℓ ) , ∀ 0 ≤ ℓ ≤ m , I ( y , Π ) ≥ I ( x , Π ) − ρ ( q , X ) , \mathcal{I}(y,\Pi)\ge\mathcal{I}(x,\Pi)-\rho(q,X), I ( y , Π ) ≥ I ( x , Π ) − ρ ( q , X ) , where, for τ ( x , Π ) = ∑ i ∈ N ∑ π ∈ Π x i ( π ) ⋅ τ i ( π ) \tau(x,\Pi)=\sum_{i\in N}\sum_{\pi\in\Pi}x_i(\pi)\cdot\tau_i(\pi) τ ( x , Π ) = ∑ i ∈ N ∑ π ∈ Π x i ( π ) ⋅ τ i ( π ) X = ∣ { ( i , π ) : 0 < x i ( π ) < 1 } ∣ X=|\{(i,\pi):0<x_i(\pi)<1\}| X = ∣ {( i , π ) : 0 < x i ( π ) < 1 } ∣
ρ ( q , X ) = 8 + 4 log λ max { 1 , 1 δ h max T max X τ ( x , Π ) } + 8 q log X . \rho(q,X)=8+4\log_{\lambda}\max\left\{1,\frac{1}{\delta}h_{\max}T_{\max}X\tau(x,\Pi)\right\}+8q\log X. ρ ( q , X ) = 8 + 4 log λ max { 1 , δ 1 h m a x T m a x X τ ( x , Π ) } + 8 q log X . Here h max : = max i ∈ N h i h_{\max}:=\max_{i\in N}h_i h m a x := max i ∈ N h i T max : = max i ∈ N T i T_{\max}:=\max_{i\in N}T_i T m a x := max i ∈ N T i
Proof. Both the statement and the proof of this theorem are very similar to Theorem A.1 . There are three key differences. First, instead of one tail load constraint, we have m + 1 m+1 m + 1 0 0 0 δ ∈ ( 0 , 1 ] \delta\in(0,1] δ ∈ ( 0 , 1 ] λ \lambda λ I \mathcal{I} I X X X π ∈ Π \pi\in\Pi π ∈ Π q q q Theorem A.1 the bound was h = O ( n 2 ) h=O(n^2) h = O ( n 2 ) Theorem A.1 , we had to put h h h O ( log ( h X ) ) = O ( log n ) O(\log(hX))=O(\log n) O ( log ( h X )) = O ( log n ) O ( q ) O(q) O ( q ) O ( log X ) O(\log X) O ( log X )
Assume X ≥ 2 X\ge2 X ≥ 2 x x x Theorem A.1 , we will slightly modify the starting solution x x x x i ( π ) x_i(\pi) x i ( π ) π ≠ ⊥ \pi\ne\perp π = ⊥ ( i , π ) (i,\pi) ( i , π ) π ≠ ⊥ \pi\ne\perp π = ⊥ 0 < x i ( π ) < 1 / X 0<x_i(\pi)<1/X 0 < x i ( π ) < 1/ X x i ( π ) x_i(\pi) x i ( π ) x i ( ⊥ ) x_i(\perp) x i ( ⊥ ) x 1 x^1 x 1 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m
H ( x 1 , Π , ℓ ) ≤ H ( x , Π , ℓ ) and T ( x 1 , Π , ℓ ) ≤ T ( x , Π , ℓ ) . \mathcal{H}(x^1,\Pi,\ell)\le\mathcal{H}(x,\Pi,\ell)
\qquad\text{and}\qquad
\mathcal{T}(x^1,\Pi,\ell)\le\mathcal{T}(x,\Pi,\ell). H ( x 1 , Π , ℓ ) ≤ H ( x , Π , ℓ ) and T ( x 1 , Π , ℓ ) ≤ T ( x , Π , ℓ ) . Since we cannot remove more than X X X 1 / X 1/X 1/ X
I ( x 1 , Π ) = ∑ i ∈ N ∑ π ≠ ⊥ x i 1 ( π ) = ∑ i ∈ N ∑ π ≠ ⊥ x i ( π ) − ∑ ( i , π ) 0 < x i ( π ) < 1 / X x i ( π ) ≥ I ( x , Π ) − 1. \mathcal{I}(x^1,\Pi)=\sum_{i\in N}\sum_{\pi\ne\perp}x_i^1(\pi)
=\sum_{i\in N}\sum_{\pi\ne\perp}x_i(\pi)
-\sum_{\substack{(i,\pi)\\0<x_i(\pi)<1/X}}x_i(\pi)
\ge\mathcal{I}(x,\Pi)-1. I ( x 1 , Π ) = i ∈ N ∑ π = ⊥ ∑ x i 1 ( π ) = i ∈ N ∑ π = ⊥ ∑ x i ( π ) − ( i , π ) 0 < x i ( π ) < 1/ X ∑ x i ( π ) ≥ I ( x , Π ) − 1. Let X 1 : = { ( i , π ) : 0 < x i 1 ( π ) < 1 } \mathcal{X}_1:=\{(i,\pi):0<x_i^1(\pi)<1\} X 1 := {( i , π ) : 0 < x i 1 ( π ) < 1 } X 1 : = ∣ X 1 ∣ X_1:=|\mathcal{X}_1| X 1 := ∣ X 1 ∣ X 1 ≤ X X_1\le X X 1 ≤ X 0 < x i 1 ( ⊥ ) < 1 0<x_i^1(\perp)<1 0 < x i 1 ( ⊥ ) < 1 i ∈ N i\in N i ∈ N x i ( π ) = 0 x_i(\pi)=0 x i ( π ) = 0 π ≠ ⊥ \pi\ne\perp π = ⊥ 0 < x i ( π ) < 1 0<x_i(\pi)<1 0 < x i ( π ) < 1
Replacing x x x x 1 x^1 x 1 T ( x 1 , Π , ℓ ) \mathcal{T}(x^1,\Pi,\ell) T ( x 1 , Π , ℓ ) 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m
τ i ( π ) : = { Δ ( π ) / T i , π ≠ ⊥ and T i > 0 ; 0 , otherwise . \tau_i(\pi):=
\begin{cases}
\Delta(\pi)/T_i, & \pi\ne\perp\text{ and }T_i>0;\\
0, & \text{otherwise}.
\end{cases} τ i ( π ) := { Δ ( π ) / T i , 0 , π = ⊥ and T i > 0 ; otherwise . By assumption, for every π ∈ Π − ⊥ \pi\in\Pi-\perp π ∈ Π − ⊥ Δ ( π ) = 0 \Delta(\pi)=0 Δ ( π ) = 0 Δ ( π ) ≥ δ > 0 \Delta(\pi)\ge\delta>0 Δ ( π ) ≥ δ > 0 T max : = max i ∈ N T i T_{\max}:=\max_{i\in N}T_i T m a x := max i ∈ N T i T max = 0 T_{\max}=0 T m a x = 0 τ i ( π ) \tau_i(\pi) τ i ( π ) 0 0 0
τ i ( π ) > 0 ⟹ τ i ( π ) ≥ δ T max . \tau_i(\pi)>0\quad\Longrightarrow\quad\tau_i(\pi)\ge\frac{\delta}{T_{\max}}. τ i ( π ) > 0 ⟹ τ i ( π ) ≥ T m a x δ . If T ( x 1 , Π , ℓ ) > 0 \mathcal{T}(x^1,\Pi,\ell)>0 T ( x 1 , Π , ℓ ) > 0 x i 1 ( π ) x_i^1(\pi) x i 1 ( π ) τ i ( π ) \tau_i(\pi) τ i ( π ) ( ℓ − h i ) + (\ell-h_i)_+ ( ℓ − h i ) +
T ( x 1 , Π , ℓ ) > 0 ⟹ T ( x 1 , Π , ℓ ) = ∑ i ∈ N ∑ π ≠ ⊥ x i 1 ( π ) ⋅ τ i ( π ) ( ℓ − h i ) + ≥ δ T max X = : θ . \mathcal{T}(x^1,\Pi,\ell)>0\quad\Longrightarrow\quad
\mathcal{T}(x^1,\Pi,\ell)=\sum_{i\in N}\sum_{\pi\ne\perp}x_i^1(\pi)\cdot\tau_i(\pi)(\ell-h_i)_+
\ge\frac{\delta}{T_{\max}X}=:\theta. T ( x 1 , Π , ℓ ) > 0 ⟹ T ( x 1 , Π , ℓ ) = i ∈ N ∑ π = ⊥ ∑ x i 1 ( π ) ⋅ τ i ( π ) ( ℓ − h i ) + ≥ T m a x X δ =: θ . In order to preserve the tail load constraints (or at least, not violate them by too much), we will do a trick similar to Lemma A.2. Specifically, before we even start rounding, we will create a set of tail checkpoints . That is, a subset of prefixes r r r T ( x k + 1 , Π , r ) ≤ T ( x k , Π , r ) \mathcal{T}(x^{k+1},\Pi,r)\le\mathcal{T}(x^k,\Pi,r) T ( x k + 1 , Π , r ) ≤ T ( x k , Π , r ) k ≥ 1 k\ge1 k ≥ 1 λ \lambda λ
We do it as follows. Let h max : = max i ∈ N h i h_{\max}:=\max_{i\in N}h_i h m a x := max i ∈ N h i t ≥ 0 t\ge0 t ≥ 0 0 ≤ r ≤ h max 0\le r\le h_{\max} 0 ≤ r ≤ h m a x T ( x 1 , Π , r ) / θ \mathcal{T}(x^1,\Pi,r)/\theta T ( x 1 , Π , r ) / θ [ λ t , λ t + 1 ) [\lambda^t,\lambda^{t+1}) [ λ t , λ t + 1 )
R t : = { r ∈ { 0 , 1 , … , h max } : θ λ t ≤ T ( x 1 , Π , r ) < θ λ t + 1 } , \mathcal{R}_t:=\left\{r\in\{0,1,\ldots,h_{\max}\}:\theta\lambda^t\le\mathcal{T}(x^1,\Pi,r)<\theta\lambda^{t+1}\right\}, R t := { r ∈ { 0 , 1 , … , h m a x } : θ λ t ≤ T ( x 1 , Π , r ) < θ λ t + 1 } , and take checkpoint r t = max r ∈ R t r r_t=\max_{r\in\mathcal{R}_t}r r t = max r ∈ R t r R \mathcal{R} R R : = ∣ R ∣ R:=|\mathcal{R}| R := ∣ R ∣
We will now show an analogue of Lemma A.2: in order to obtain the desired guarantees for y y y T ( y , Π , ℓ ) \mathcal{T}(y,\Pi,\ell) T ( y , Π , ℓ ) r ∈ R r\in\mathcal{R} r ∈ R
Lemma B.1. Suppose that a solution z z z z i ( π ) > 0 z_i(\pi)>0 z i ( π ) > 0 π ≠ ⊥ \pi\ne\bot π = ⊥ x i 1 ( π ) > 0 x_i^1(\pi)>0 x i 1 ( π ) > 0 τ ( z , Π ) ≤ τ ( x 1 , Π ) \tau(z,\Pi)\le\tau(x^1,\Pi) τ ( z , Π ) ≤ τ ( x 1 , Π ) T ( z , Π , r ) ≤ T ( x 1 , Π , r ) \mathcal{T}(z,\Pi,r)\le\mathcal{T}(x^1,\Pi,r) T ( z , Π , r ) ≤ T ( x 1 , Π , r ) r ∈ R r\in\mathcal{R} r ∈ R 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m
T ( z , Π , ℓ ) ≤ λ T ( x 1 , Π , ℓ ) , \mathcal{T}(z,\Pi,\ell)\le\lambda\mathcal{T}(x^1,\Pi,\ell), T ( z , Π , ℓ ) ≤ λ T ( x 1 , Π , ℓ ) , and in fact T ( z , Π , ℓ ) ≤ T ( x 1 , Π , ℓ ) \mathcal{T}(z,\Pi,\ell)\le\mathcal{T}(x^1,\Pi,\ell) T ( z , Π , ℓ ) ≤ T ( x 1 , Π , ℓ ) ℓ ≥ h max \ell\ge h_{\max} ℓ ≥ h m a x R = ∣ R ∣ R=|\mathcal{R}| R = ∣ R ∣
R ≤ 1 + log λ max { 1 , h max τ ( x , Π ) θ } . R\le1+\log_{\lambda}\max\left\{1,\frac{h_{\max}\tau(x,\Pi)}{\theta}\right\}. R ≤ 1 + log λ max { 1 , θ h m a x τ ( x , Π ) } . Proof. If h max = 0 h_{\max}=0 h m a x = 0 τ ( z , Π ) ≤ τ ( x 1 , Π ) \tau(z,\Pi)\le\tau(x^1,\Pi) τ ( z , Π ) ≤ τ ( x 1 , Π ) h max > 0 h_{\max}>0 h m a x > 0
Suppose that T ( x 1 , Π , ℓ ) = 0 \mathcal{T}(x^1,\Pi,\ell)=0 T ( x 1 , Π , ℓ ) = 0 π ∈ Π \pi\in\Pi π ∈ Π x i 1 ( π ) > 0 x_i^1(\pi)>0 x i 1 ( π ) > 0 z i ( π ) > 0 z_i(\pi)>0 z i ( π ) > 0 π ≠ ⊥ \pi\ne\bot π = ⊥ x i 1 ( π ) > 0 x_i^1(\pi)>0 x i 1 ( π ) > 0 T ( z , Π , ℓ ) = 0 \mathcal{T}(z,\Pi,\ell)=0 T ( z , Π , ℓ ) = 0
Next, consider ℓ < h max \ell<h_{\max} ℓ < h m a x T ( x 1 , Π , ℓ ) > 0 \mathcal{T}(x^1,\Pi,\ell)>0 T ( x 1 , Π , ℓ ) > 0 ℓ \ell ℓ R t \mathcal{R}_t R t t ≥ 0 t\ge0 t ≥ 0 θ λ t ≤ T ( x 1 , Π , ℓ ) < θ λ t + 1 \theta\lambda^t\le\mathcal{T}(x^1,\Pi,\ell)<\theta\lambda^{t+1} θ λ t ≤ T ( x 1 , Π , ℓ ) < θ λ t + 1 r t = max r ∈ R t r r_t=\max_{r\in\mathcal{R}_t}r r t = max r ∈ R t r ℓ ≤ r t \ell\le r_t ℓ ≤ r t
T ( z , Π , ℓ ) ≤ T ( z , Π , r t ) ≤ T ( x 1 , Π , r t ) ≤ θ λ t + 1 ≤ λ T ( x 1 , Π , ℓ ) . \mathcal{T}(z,\Pi,\ell)\le\mathcal{T}(z,\Pi,r_t)\le\mathcal{T}(x^1,\Pi,r_t)\le\theta\lambda^{t+1}\le\lambda\mathcal{T}(x^1,\Pi,\ell). T ( z , Π , ℓ ) ≤ T ( z , Π , r t ) ≤ T ( x 1 , Π , r t ) ≤ θ λ t + 1 ≤ λ T ( x 1 , Π , ℓ ) . Note that if T ( x 1 , Π , h max ) > 0 \mathcal{T}(x^1,\Pi,h_{\max})>0 T ( x 1 , Π , h m a x ) > 0 h max ∈ R h_{\max}\in\mathcal{R} h m a x ∈ R T ( z , Π , h max ) ≤ T ( x 1 , Π , h max ) \mathcal{T}(z,\Pi,h_{\max})\le\mathcal{T}(x^1,\Pi,h_{\max}) T ( z , Π , h m a x ) ≤ T ( x 1 , Π , h m a x ) T ( x 1 , Π , h max ) = 0 \mathcal{T}(x^1,\Pi,h_{\max})=0 T ( x 1 , Π , h m a x ) = 0 T ( z , Π , h max ) = 0 \mathcal{T}(z,\Pi,h_{\max})=0 T ( z , Π , h m a x ) = 0 ℓ ≥ h max \ell\ge h_{\max} ℓ ≥ h m a x
T ( z , Π , ℓ ) = ∑ i ∈ N τ i ( z i , Π ) ⋅ ( ℓ − h i ) = ∑ i ∈ N τ i ( z i , Π ) ⋅ ( h max − h i ) + ∑ i ∈ N τ i ( z i , Π ) ⋅ ( ℓ − h max ) = T ( z , Π , h max ) + ( ℓ − h max ) τ ( z , Π ) ≤ T ( x 1 , Π , h max ) + ( ℓ − h max ) τ ( x 1 , Π ) = T ( x 1 , Π , ℓ ) . \begin{aligned}
\mathcal{T}(z,\Pi,\ell)
&=\sum_{i\in N}\tau_i(z_i,\Pi)\cdot(\ell-h_i)\\
&=\sum_{i\in N}\tau_i(z_i,\Pi)\cdot(h_{\max}-h_i)+\sum_{i\in N}\tau_i(z_i,\Pi)\cdot(\ell-h_{\max})\\
&=\mathcal{T}(z,\Pi,h_{\max})+(\ell-h_{\max})\tau(z,\Pi)\\
&\le\mathcal{T}(x^1,\Pi,h_{\max})+(\ell-h_{\max})\tau(x^1,\Pi)=\mathcal{T}(x^1,\Pi,\ell).
\end{aligned} T ( z , Π , ℓ ) = i ∈ N ∑ τ i ( z i , Π ) ⋅ ( ℓ − h i ) = i ∈ N ∑ τ i ( z i , Π ) ⋅ ( h m a x − h i ) + i ∈ N ∑ τ i ( z i , Π ) ⋅ ( ℓ − h m a x ) = T ( z , Π , h m a x ) + ( ℓ − h m a x ) τ ( z , Π ) ≤ T ( x 1 , Π , h m a x ) + ( ℓ − h m a x ) τ ( x 1 , Π ) = T ( x 1 , Π , ℓ ) . We now bound R R R τ ( x 1 , Π ) ≤ τ ( x , Π ) \tau(x^1,\Pi)\le\tau(x,\Pi) τ ( x 1 , Π ) ≤ τ ( x , Π ) ⊥ \bot ⊥ ℓ ≤ h max \ell\le h_{\max} ℓ ≤ h m a x
T ( x 1 , Π , ℓ ) = ∑ i ∈ N τ i ( x i 1 , Π ) ⋅ ( ℓ − h i ) + ≤ ℓ ⋅ τ ( x 1 , Π ) ≤ h max τ ( x , Π ) . \mathcal{T}(x^1,\Pi,\ell)=\sum_{i\in N}\tau_i(x_i^1,\Pi)\cdot(\ell-h_i)_+\le\ell\cdot\tau(x^1,\Pi)\le h_{\max}\tau(x,\Pi). T ( x 1 , Π , ℓ ) = i ∈ N ∑ τ i ( x i 1 , Π ) ⋅ ( ℓ − h i ) + ≤ ℓ ⋅ τ ( x 1 , Π ) ≤ h m a x τ ( x , Π ) . If h max τ ( x , Π ) < θ h_{\max}\tau(x,\Pi)<\theta h m a x τ ( x , Π ) < θ R t \mathcal{R}_t R t R = 0 R=0 R = 0 t ≥ 0 t\ge0 t ≥ 0 R t ≠ ∅ \mathcal{R}_t\ne\varnothing R t = ∅
θ λ t ≤ h max τ ( x , Π ) ⟹ t ≤ log λ max { 1 , h max τ ( x , Π ) θ } . \theta\lambda^t\le h_{\max}\tau(x,\Pi)\quad\Longrightarrow\quad t\le\log_{\lambda}\max\left\{1,\frac{h_{\max}\tau(x,\Pi)}{\theta}\right\}. θ λ t ≤ h m a x τ ( x , Π ) ⟹ t ≤ log λ max { 1 , θ h m a x τ ( x , Π ) } . There is at most one tail checkpoint for each such t ≥ 0 t\ge0 t ≥ 0 □ \square □
We now proceed to actual rounding. As in Theorem A.1 , we start with a solution x 1 x^1 x 1 y y y ⊥ \bot ⊥ x k x^k x k i ∈ N i\in N i ∈ N x i k ( π ) ≥ 0 x_i^k(\pi)\ge0 x i k ( π ) ≥ 0 π ∈ Π \pi\in\Pi π ∈ Π ∑ π ∈ Π x i k ( π ) = 1 \sum_{\pi\in\Pi}x_i^k(\pi)=1 ∑ π ∈ Π x i k ( π ) = 1 X k : = { ( i , π ) : 0 < x i k ( π ) < 1 } \mathcal{X}_k:=\{(i,\pi):0<x_i^k(\pi)<1\} X k := {( i , π ) : 0 < x i k ( π ) < 1 } x k x^k x k X k : = ∣ X k ∣ X_k:=|\mathcal{X}_k| X k := ∣ X k ∣
Suppose that X k > 8 ( R + 1 ) X_k>8(R+1) X k > 8 ( R + 1 ) R \mathcal{R} R k k k k k k head checkpoints ℓ \ell ℓ H ( x k + 1 , Π , ℓ ) ≤ H ( x k , Π , ℓ ) \mathcal{H}(x^{k+1},\Pi,\ell)\le\mathcal{H}(x^k,\Pi,\ell) H ( x k + 1 , Π , ℓ ) ≤ H ( x k , Π , ℓ ) O ( log q X ) O(\log qX) O ( log q X ) Theorem A.1 , we will save a log X \log X log X O ( q ) O(q) O ( q )
We create additional slackness in every prefix 0 ≤ ℓ ≤ m 0 \le\ell\le m 0 ≤ ℓ ≤ m intermediate solution x ~ k \tilde{x}^{k} x ~ k ( i , π ) ∈ X k (i,\pi) \in\mathcal{X}_{k} ( i , π ) ∈ X k π ∉ { ∅ , ⊥ } \pi\notin\{\varnothing,\bot\} π ∈ / { ∅ , ⊥ }
b i ( π ) : = min α ∈ π m i ( α ) b_i(\pi) := \min_{\alpha\in\pi} m_i(\alpha) b i ( π ) := α ∈ π min m i ( α ) be the earliest breakpoint of agent i i i π \pi π ( i , π ) ∈ X k (i,\pi) \in\mathcal{X}_{k} ( i , π ) ∈ X k b i ( π ) b_i(\pi) b i ( π ) x i k ( π ) x_i^k(\pi) x i k ( π ) x i k ( ⊥ ) x_i^k(\bot) x i k ( ⊥ ) 8 q 8q 8 q ⊥ \bot ⊥ 0 0 0 x i k ( π ) x_i^k(\pi) x i k ( π ) x ~ k \tilde{x}^{k} x ~ k X ~ k : = { ( i , π ) : 0 < x ~ i k ( π ) < 1 } \widetilde{\mathcal{X}}_{k} := \{(i,\pi) : 0 < \tilde{x}_i^k(\pi) < 1\} X k := {( i , π ) : 0 < x ~ i k ( π ) < 1 } X ~ k : = ∣ X ~ k ∣ \widetilde{X}_{k} := |\widetilde{\mathcal{X}}_{k}| X k := ∣ X k ∣
As in the proof of Theorem A.1 , for ( i , π ) ∈ X ~ k (i,\pi) \in\widetilde{\mathcal{X}}_{k} ( i , π ) ∈ X k π ≠ ⊥ \pi\ne\bot π = ⊥ i i i α ∈ π \alpha\in\pi α ∈ π M k \mathcal{M}_{k} M k
\mathcal{M}_{k} := \mathbin{\mathaccent\cdot\cup}_{\substack{(i,\pi)\in\widetilde{\mathcal{X}}_{k}\\ \pi\ne\bot}} \{m_i(\alpha) : \alpha\in\pi\}.
As before, for − 1 ≤ ℓ ≤ m -1 \le\ell\le m − 1 ≤ ℓ ≤ m
P k ( ℓ ) = ∑ ( i , π ) ∈ X ~ k P i ( π , ℓ ) > 0 \mathcal{P}_{k}(\ell) = \sum_{(i,\pi)\in\widetilde{\mathcal{X}}_{k}} \mathcal{P}_{i}(\pi,\ell) > 0 P k ( ℓ ) = ( i , π ) ∈ X k ∑ P i ( π , ℓ ) > 0 the total number of breakpoints in M k \mathcal{M}_{k} M k ℓ \ell ℓ P k ( − 1 ) = 0 P_k(-1) = 0 P k ( − 1 ) = 0
Lemma B.2. It holds I ( x ~ k , Π ) ≥ I ( x k , Π ) − 8 q \mathcal{I}(\tilde{x}^{k},\Pi) \ge\mathcal{I}(x^{k},\Pi)-8q I ( x ~ k , Π ) ≥ I ( x k , Π ) − 8 q τ ( x ~ k , Π ) ≤ τ ( x k , Π ) \tau(\tilde{x}^{k},\Pi) \le\tau(x^{k},\Pi) τ ( x ~ k , Π ) ≤ τ ( x k , Π ) T ( x ~ k , Π , ℓ ) ≤ T ( x k , Π , ℓ ) \mathcal{T}(\tilde{x}^{k},\Pi,\ell) \le\mathcal{T}(x^{k},\Pi,\ell) T ( x ~ k , Π , ℓ ) ≤ T ( x k , Π , ℓ ) H ( x ~ k , Π , ℓ ) ≤ H ( x k , Π , ℓ ) \mathcal{H}(\tilde{x}^{k},\Pi,\ell) \le\mathcal{H}(x^{k},\Pi,\ell) H ( x ~ k , Π , ℓ ) ≤ H ( x k , Π , ℓ ) 0 ≤ ℓ ≤ m 0 \le\ell\le m 0 ≤ ℓ ≤ m P k ( ℓ ) > 0 P_k(\ell) > 0 P k ( ℓ ) > 0
H ( x ~ k , Π , ℓ ) + 8 q ≤ H ( x k , Π , ℓ ) . \mathcal{H}(\tilde{x}^{k},\Pi,\ell) + 8q \le\mathcal{H}(x^{k},\Pi,\ell). H ( x ~ k , Π , ℓ ) + 8 q ≤ H ( x k , Π , ℓ ) . Proof. Every claim except for the last bound on head prefix loads is immediate. Let ℓ \ell ℓ P k ( ℓ ) > 0 P_k(\ell) > 0 P k ( ℓ ) > 0 ( i ∗ , π ∗ ) ∈ X ~ k (i_*,\pi_*) \in\widetilde{\mathcal{X}}_{k} ( i ∗ , π ∗ ) ∈ X k P i ∗ ( π ∗ , ℓ ) > 0 \mathcal{P}_{i_*}(\pi_*,\ell) > 0 P i ∗ ( π ∗ , ℓ ) > 0 π ∗ \pi_* π ∗ b i ∗ ( π ∗ ) ≤ ℓ b_{i_*}(\pi_*) \le\ell b i ∗ ( π ∗ ) ≤ ℓ x ~ i ∗ k ( π ∗ ) > 0 \tilde{x}_{i_*}^k(\pi_*) > 0 x ~ i ∗ k ( π ∗ ) > 0 x i ∗ k ( π ∗ ) > 0 x_{i_*}^k(\pi_*) > 0 x i ∗ k ( π ∗ ) > 0 ( i , π ) ∈ X k (i,\pi) \in\mathcal{X}_{k} ( i , π ) ∈ X k ⊥ \bot ⊥ ( i ∗ , π ∗ ) (i_*,\pi_*) ( i ∗ , π ∗ ) ( i ∗ , π ∗ ) (i_*,\pi_*) ( i ∗ , π ∗ ) ( i , π ) (i,\pi) ( i , π ) x i k ( π ) − x ~ i k ( π ) > 0 x_i^k(\pi)-\tilde{x}_i^k(\pi)>0 x i k ( π ) − x ~ i k ( π ) > 0 b i ( π ) ≤ b i ∗ ( π ∗ ) ≤ ℓ b_i(\pi) \le b_{i_*}(\pi_*) \le\ell b i ( π ) ≤ b i ∗ ( π ∗ ) ≤ ℓ P i ( π , ℓ ) ≥ 1 \mathcal{P}_i(\pi,\ell) \ge1 P i ( π , ℓ ) ≥ 1 ( i , π ) (i,\pi) ( i , π )
H ( x k , Π , ℓ ) − H ( x ~ k , Π , ℓ ) = ∑ i ∈ N ∑ π ≠ ⊥ ( x i k ( π ) − x ~ i k ( π ) ) ⋅ P i ( π , ℓ ) ≥ 8 q . \mathcal{H}(x^k,\Pi,\ell)-\mathcal{H}(\tilde{x}^k,\Pi,\ell)=\sum_{i\in N}\sum_{\pi\ne\bot}(x_i^k(\pi)-\tilde{x}_i^k(\pi))\cdot\mathcal{P}_i(\pi,\ell)\ge8q. H ( x k , Π , ℓ ) − H ( x ~ k , Π , ℓ ) = i ∈ N ∑ π = ⊥ ∑ ( x i k ( π ) − x ~ i k ( π )) ⋅ P i ( π , ℓ ) ≥ 8 q . It is at least 8 q 8q 8 q x ~ i ∗ k ( π ∗ ) > 0 \tilde{x}_{i_*}^k(\pi_*)>0 x ~ i ∗ k ( π ∗ ) > 0 □ \square □
We now construct head checkpoints . Sort M k \mathcal{M}_{k} M k 8 q 8q 8 q M k \mathcal{M}_{k} M k L k \mathcal{L}_{k} L k
Lemma B.3. The total number of head checkpoints at phase k k k ∣ L k ∣ ≤ X k / 8 |\mathcal{L}_{k}| \le X_{k}/8 ∣ L k ∣ ≤ X k /8 z z z z i ( π ) = x ~ i k ( π ) z_i(\pi)=\tilde{x}_i^k(\pi) z i ( π ) = x ~ i k ( π ) ( i , π ) (i,\pi) ( i , π ) x ~ i k ( π ) ∈ { 0 , 1 } \tilde{x}_i^k(\pi)\in\{0,1\} x ~ i k ( π ) ∈ { 0 , 1 } Suppose that for every ℓ k ∈ L k \ell_k\in\mathcal{L}_{k} ℓ k ∈ L k H ( z , Π , ℓ k ) ≤ H ( x ~ k , Π , ℓ k ) \mathcal{H}(z,\Pi,\ell_k)\le\mathcal{H}(\tilde{x}^{k},\Pi,\ell_k) H ( z , Π , ℓ k ) ≤ H ( x ~ k , Π , ℓ k ) 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m
H ( z , Π , ℓ ) ≤ H ( x ~ k , Π , ℓ ) + 8 q 1 { P k ( ℓ ) > 0 } . \mathcal{H}(z,\Pi,\ell)\le\mathcal{H}(\tilde{x}^{k},\Pi,\ell)+8q\mathbf{1}\{P_k(\ell)>0\}. H ( z , Π , ℓ ) ≤ H ( x ~ k , Π , ℓ ) + 8 q 1 { P k ( ℓ ) > 0 } . Proof. This is an analogue of Lemma A.2 for heads. By construction, every fractional coordinate in x ~ k \tilde{x}^{k} x ~ k x k x^{k} x k ∣ π ∣ ≤ q |\pi|\le q ∣ π ∣ ≤ q π ∈ Π \pi\in\Pi π ∈ Π
∣ M k ∣ = ∑ ( i , π ) ∈ X ~ k π ≠ ⊥ ∣ π ∣ ≤ q X k ⟹ ∣ L k ∣ ≤ ⌈ ∣ M k ∣ 8 q ⌉ ≤ X k 8 . |\mathcal{M}_{k}|=\sum_{\substack{(i,\pi)\in\widetilde{\mathcal{X}}_{k}\\ \pi\ne\bot}}|\pi|\le qX_{k}\quad\Longrightarrow\quad|\mathcal{L}_{k}|\le\left\lceil\frac{|\mathcal{M}_{k}|}{8q}\right\rceil\le\frac{X_{k}}{8}. ∣ M k ∣ = ( i , π ) ∈ X k π = ⊥ ∑ ∣ π ∣ ≤ q X k ⟹ ∣ L k ∣ ≤ ⌈ 8 q ∣ M k ∣ ⌉ ≤ 8 X k . Next, let 0 ≤ ℓ ≤ m 0 \le\ell\le m 0 ≤ ℓ ≤ m P k ( ℓ ) = 0 P_k(\ell)=0 P k ( ℓ ) = 0 x ~ k \tilde{x}^k x ~ k 0 0 0 H ( x ~ k , Π , ℓ ) \mathcal{H}(\tilde{x}^k,\Pi,\ell) H ( x ~ k , Π , ℓ ) z z z x ~ k \tilde{x}^k x ~ k X k X_k X k H ( z , Π , ℓ ) = H ( x ~ k , Π , ℓ ) \mathcal{H}(z,\Pi,\ell)=\mathcal{H}(\tilde{x}^k,\Pi,\ell) H ( z , Π , ℓ ) = H ( x ~ k , Π , ℓ )
If P k ( ℓ ) > 0 P_k(\ell)>0 P k ( ℓ ) > 0 ℓ k ∈ L k \ell_k \in L_k ℓ k ∈ L k ℓ \ell ℓ ℓ k ≤ ℓ \ell_k \le\ell ℓ k ≤ ℓ ℓ k = − 1 \ell_k=-1 ℓ k = − 1
H ( z , Π , ℓ ) − H ( x ~ k , Π , ℓ ) ≤ P k ( ℓ ) − P k ( ℓ k ) . \mathcal{H}(z,\Pi,\ell)-\mathcal{H}(\tilde{x}^k,\Pi,\ell)\le P_k(\ell)-P_k(\ell_k). H ( z , Π , ℓ ) − H ( x ~ k , Π , ℓ ) ≤ P k ( ℓ ) − P k ( ℓ k ) . If ℓ k = ℓ \ell_k=\ell ℓ k = ℓ P k ( ℓ ) − P k ( ℓ k ) = 0 P_k(\ell)-P_k(\ell_k)=0 P k ( ℓ ) − P k ( ℓ k ) = 0 L k L_k L k 8 q 8q 8 q M k M_k M k ( ℓ k , ℓ ] (\ell_k,\ell] ( ℓ k , ℓ ] P k ( ℓ ) − P k ( ℓ k ) ≤ 8 q P_k(\ell)-P_k(\ell_k)\le8q P k ( ℓ ) − P k ( ℓ k ) ≤ 8 q
It follows from Lemma B.1, Lemma B.2 and Lemma B.3 that, in order to preserve all head and tail load constraints at every prefix 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m k k k R \mathcal{R} R L k L_k L k τ ( x k + 1 , Π ) ≤ τ ( x k , Π ) \tau(x^{k+1},\Pi)\le\tau(x^k,\Pi) τ ( x k + 1 , Π ) ≤ τ ( x k , Π )
Similar to Theorem 2.2, we construct a polytope (P):
∑ π ∈ Π z i ( π ) = 1 , ∀ i ∈ N , z i ( π ) = x ~ i k ( π ) , ∀ ( i , π ) : x ~ i k ( π ) ∈ { 0 , 1 } , τ ( z , Π ) ≤ τ ( x ~ k , Π ) , T ( z , Π , r ) ≤ T ( x ~ k , Π , r ) , ∀ r ∈ R , H ( z , Π , ℓ k ) ≤ H ( x ~ k , Π , ℓ k ) , ∀ ℓ k ∈ L k , 0 ≤ z i ( π ) ≤ 1 , ∀ i ∈ N , π ∈ Π . \begin{aligned}
\sum_{\pi\in\Pi} z_i(\pi) &= 1, && \forall i\in N,\\
z_i(\pi) &= \tilde{x}_i^k(\pi), && \forall(i,\pi):\tilde{x}_i^k(\pi)\in\{0,1\},\\
\tau(z,\Pi) &\le\tau(\tilde{x}^k,\Pi),\\
\mathcal{T}(z,\Pi,r) &\le\mathcal{T}(\tilde{x}^k,\Pi,r), && \forall r\in\mathcal{R},\\
\mathcal{H}(z,\Pi,\ell_k) &\le\mathcal{H}(\tilde{x}^k,\Pi,\ell_k), && \forall\ell_k\in L_k,\\
0\le z_i(\pi)&\le1, && \forall i\in N,\ \pi\in\Pi.
\end{aligned} π ∈ Π ∑ z i ( π ) z i ( π ) τ ( z , Π ) T ( z , Π , r ) H ( z , Π , ℓ k ) 0 ≤ z i ( π ) = 1 , = x ~ i k ( π ) , ≤ τ ( x ~ k , Π ) , ≤ T ( x ~ k , Π , r ) , ≤ H ( x ~ k , Π , ℓ k ) , ≤ 1 , ∀ i ∈ N , ∀ ( i , π ) : x ~ i k ( π ) ∈ { 0 , 1 } , ∀ r ∈ R , ∀ ℓ k ∈ L k , ∀ i ∈ N , π ∈ Π. Clearly, z = x ~ k z=\tilde{x}^k z = x ~ k z ∗ z^* z ∗
I ( z ∗ , Π ) = ∑ i ∈ N ∑ π ≠ ⊥ z i ∗ ( π ) . \mathcal{I}(z^*,\Pi)=\sum_{i\in N}\sum_{\pi\ne\bot}z_i^*(\pi). I ( z ∗ , Π ) = i ∈ N ∑ π = ⊥ ∑ z i ∗ ( π ) . It exists, since (P) is nonempty and compact. By maximality, I ( z ∗ , Π ) ≥ I ( x ~ k , Π ) \mathcal{I}(z^*,\Pi)\ge\mathcal{I}(\tilde{x}^k,\Pi) I ( z ∗ , Π ) ≥ I ( x ~ k , Π )
Lemma B.4. Let z ∗ z^* z ∗ z ∗ z^* z ∗ X k / 2 X_k/2 X k /2
Proof. The proof is completely analogous to the proof of Lemma A.3. If Z ∗ Z^* Z ∗ z ∗ z^* z ∗ Z ∗ / 2 Z^*/2 Z ∗ /2 i ∈ N i\in N i ∈ N z i ∗ ( π ) z_i^*(\pi) z i ∗ ( π ) π ∈ Π \pi\in\Pi π ∈ Π
Suppose that Z ∗ > X k / 2 Z^*>X_k/2 Z ∗ > X k /2 z ∗ z^* z ∗ Z ∗ Z^* Z ∗ Z ∗ / 2 Z^*/2 Z ∗ /2 ∑ π ∈ Π z i ( π ) = 1 \sum_{\pi\in\Pi}z_i(\pi)=1 ∑ π ∈ Π z i ( π ) = 1 i ∈ N i\in N i ∈ N R R R ∣ L k ∣ ≤ X k / 8 |L_k|\le X_k/8 ∣ L k ∣ ≤ X k /8 τ ( z , Π ) ≤ τ ( x k , Π ) \tau(z,\Pi)\le\tau(x^k,\Pi) τ ( z , Π ) ≤ τ ( x k , Π ) X k > 8 ( R + 1 ) X_k>8(R+1) X k > 8 ( R + 1 )
Z ∗ 2 + R + X k 8 + 1 ≤ Z ∗ 2 + X k 4 < Z ∗ , \frac{Z^*}{2}+R+\frac{X_k}{8}+1\le\frac{Z^*}{2}+\frac{X_k}{4}<Z^*, 2 Z ∗ + R + 8 X k + 1 ≤ 2 Z ∗ + 4 X k < Z ∗ , where we used Z ∗ > X k / 2 Z^*>X_k/2 Z ∗ > X k /2 z ∗ z^* z ∗
We take this extreme point z ∗ z^* z ∗ x k + 1 = z ∗ x^{k+1}=z^* x k + 1 = z ∗ X k + 1 ≤ X k / 2 X_{k+1}\le X_k/2 X k + 1 ≤ X k /2 τ ( x k + 1 , Π ) ≤ τ ( x k , Π ) \tau(x^{k+1},\Pi)\le\tau(x^k,\Pi) τ ( x k + 1 , Π ) ≤ τ ( x k , Π ) I ( x k + 1 , Π ) ≥ I ( x k , Π ) − 8 q \mathcal{I}(x^{k+1},\Pi)\ge\mathcal{I}(x^k,\Pi)-8q I ( x k + 1 , Π ) ≥ I ( x k , Π ) − 8 q 0 ≤ ℓ ≤ m 0\le\ell\le m 0 ≤ ℓ ≤ m H ( x k + 1 , Π , ℓ ) ≤ H ( x k , Π , ℓ ) \mathcal{H}(x^{k+1},\Pi,\ell)\le\mathcal{H}(x^k,\Pi,\ell) H ( x k + 1 , Π , ℓ ) ≤ H ( x k , Π , ℓ ) r ∈ R r\in\mathcal{R} r ∈ R T ( x k + 1 , Π , r ) ≤ T ( x k , Π , r ) \mathcal{T}(x^{k+1},\Pi,r)\le\mathcal{T}(x^k,\Pi,r) T ( x k + 1 , Π , r ) ≤ T ( x k , Π , r )
We repeat the process until at some phase K + 1 K+1 K + 1 X K + 1 ≤ 8 ( R + 1 ) X_{K+1}\le8(R+1) X K + 1 ≤ 8 ( R + 1 ) K K K
I ( x K + 1 , Π ) ≥ I ( x 1 , Π ) − 8 q K , \mathcal{I}(x^{K+1},\Pi)\ge\mathcal{I}(x^1,\Pi)-8qK, I ( x K + 1 , Π ) ≥ I ( x 1 , Π ) − 8 q K , τ ( x K + 1 , Π ) ≤ τ ( x 1 , Π ) , \tau(x^{K+1},\Pi)\le\tau(x^1,\Pi), τ ( x K + 1 , Π ) ≤ τ ( x 1 , Π ) , H ( x K + 1 , Π , ℓ ) ≤ H ( x 1 , Π , ℓ ) , ∀ 0 ≤ ℓ ≤ m , \mathcal{H}(x^{K+1},\Pi,\ell)\le\mathcal{H}(x^1,\Pi,\ell),\qquad\forall\,0\le\ell\le m, H ( x K + 1 , Π , ℓ ) ≤ H ( x 1 , Π , ℓ ) , ∀ 0 ≤ ℓ ≤ m , T ( x K + 1 , Π , r ) ≤ T ( x 1 , Π , r ) , ∀ r ∈ R . \mathcal{T}(x^{K+1},\Pi,r)\le\mathcal{T}(x^1,\Pi,r),\qquad\forall r\in\mathcal{R}. T ( x K + 1 , Π , r ) ≤ T ( x 1 , Π , r ) , ∀ r ∈ R . In addition, at most X K + 1 / 2 ≤ 4 ( R + 1 ) X_{K+1}/2 \le4(R+1) X K + 1 /2 ≤ 4 ( R + 1 ) ⊥ \bot ⊥ y y y y y y
∀ 0 ≤ ℓ ≤ m : T ( y , Π , ℓ ) ≤ λ T ( x 1 , Π , ℓ ) ≤ λ T ( x , Π , ℓ ) . \forall0 \le\ell\le m:\quad\mathcal{T}(y,\Pi,\ell) \le\lambda\mathcal{T}(x^{1},\Pi,\ell) \le\lambda\mathcal{T}(x,\Pi,\ell). ∀0 ≤ ℓ ≤ m : T ( y , Π , ℓ ) ≤ λ T ( x 1 , Π , ℓ ) ≤ λ T ( x , Π , ℓ ) . At the same time,
I ( y , Π ) ≥ I ( x K + 1 , Π ) − 4 ( R + 1 ) ≥ I ( x 1 , Π ) − 8 q K − 4 ( R + 1 ) ≥ I ( x , Π ) − 1 − 8 q K − 4 ( R + 1 ) . \begin{aligned}
\mathcal{I}(y,\Pi) \ge\mathcal{I}(x^{K+1},\Pi) - 4(R+1) \ge\mathcal{I}(x^{1},\Pi) - 8qK - 4(R+1) \\
&\ge\mathcal{I}(x,\Pi) - 1 - 8qK - 4(R+1).
\end{aligned} I ( y , Π ) ≥ I ( x K + 1 , Π ) − 4 ( R + 1 ) ≥ I ( x 1 , Π ) − 8 q K − 4 ( R + 1 ) ≥ I ( x , Π ) − 1 − 8 q K − 4 ( R + 1 ) . The last phase K K K
8 ( R + 1 ) < X K ≤ X ⋅ 2 − ( K − 1 ) , 8(R+1) < X_K \le X\cdot2^{-(K-1)}, 8 ( R + 1 ) < X K ≤ X ⋅ 2 − ( K − 1 ) , implying K ≤ log X − 1 K \le\log X - 1 K ≤ log X − 1
1 + 8 q K + 4 ( R + 1 ) ≤ 9 + 4 log λ max { 1 , h max τ ( x , Π ) θ } + 8 q ( log X − 1 ) 1 + 8qK + 4(R+1) \le9 + 4\log_{\lambda}\max\left\{1,\frac{h_{\max}\tau(x,\Pi)}{\theta}\right\} + 8q(\log X - 1) 1 + 8 q K + 4 ( R + 1 ) ≤ 9 + 4 log λ max { 1 , θ h m a x τ ( x , Π ) } + 8 q ( log X − 1 ) ≤ 8 + 4 log λ max { 1 , 1 δ h max T max X τ ( x , Π ) } + 8 q log X . \le8 + 4\log_{\lambda}\max\left\{1,\frac{1}{\delta}h_{\max}T_{\max}X\tau(x,\Pi)\right\} + 8q\log X. ≤ 8 + 4 log λ max { 1 , δ 1 h m a x T m a x X τ ( x , Π ) } + 8 q log X . By construction, the agents i ∈ N i \in N i ∈ N y i ( π ) = 1 y_i(\pi)=1 y i ( π ) = 1 π ≠ ⊥ \pi\ne\bot π = ⊥ x i ( π ) > 0 x_i(\pi)>0 x i ( π ) > 0
Proof of the single supplier theorem
Theorem C.1 (Theorem 5.1 restated). For some global constant C t , q C_{t,q} C t , q For u ∗ ∈ U u_* \in U u ∗ ∈ U { π u ∗ , j : j ∈ [ d ] } \{\pi_{u_*,j}:j \in[d]\} { π u ∗ , j : j ∈ [ d ]} L S ⊆ { 0 , 1 , … , m } \mathcal{L}_S \subseteq\{0,1,\ldots,m\} L S ⊆ { 0 , 1 , … , m } t t t protected prefixes . There exist A 0 ⊆ A A_0 \subseteq A A 0 ⊆ A J 0 ⊆ [ d ] J_0 \subseteq[d] J 0 ⊆ [ d ] ∣ A 0 ∣ + ∣ J 0 ∣ = d |A_0|+|J_0|=d ∣ A 0 ∣ + ∣ J 0 ∣ = d
∣ J 0 ∣ ≤ C t , q ( d − ∣ A ∣ ) |J_0| \le C_{t,q}(d-|A|) ∣ J 0 ∣ ≤ C t , q ( d − ∣ A ∣ ) E E E
∑ j ∈ J 0 ∣ π u ∗ , j E ∣ ≤ q C t , q ( d − ∣ A ∣ ) ; \sum_{j\in J_0}|\pi^E_{u_*,j}| \le qC_{t,q}(d-|A|); j ∈ J 0 ∑ ∣ π u ∗ , j E ∣ ≤ q C t , q ( d − ∣ A ∣ ) ; for every protected prefix ℓ s ∈ L S \ell_s \in\mathcal{L}_S ℓ s ∈ L S E E E
∑ i ∈ A 0 P i ( π i , ℓ s ) + ∑ j ∈ J 0 P u ∗ ( π u ∗ , j , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣ . \sum_{i\in A_0}\mathcal{P}_i(\pi_i,\ell_s)+\sum_{j\in J_0}\mathcal{P}_{u_*}(\pi_{u_*,j},\ell_s)\le|E\cap[\ell_s]|. i ∈ A 0 ∑ P i ( π i , ℓ s ) + j ∈ J 0 ∑ P u ∗ ( π u ∗ , j , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣. Proof. Note that the bound on ∑ j ∈ J 0 ∣ π u ∗ , j E ∣ \sum_{j\in J_0}|\pi^E_{u_*,j}| ∑ j ∈ J 0 ∣ π u ∗ , j E ∣ ∣ J 0 ∣ ≤ C t , q ( d − ∣ A ∣ ) |J_0|\le C_{t,q}(d-|A|) ∣ J 0 ∣ ≤ C t , q ( d − ∣ A ∣ ) π u ∗ , j \pi_{u_*,j} π u ∗ , j q q q ∣ J 0 ∣ |J_0| ∣ J 0 ∣ A 0 ⊆ A A_0\subseteq A A 0 ⊆ A J 0 ⊆ [ d ] J_0\subseteq[d] J 0 ⊆ [ d ]
∣ A 0 ∣ + ∣ J 0 ∣ = d |A_0|+|J_0|=d ∣ A 0 ∣ + ∣ J 0 ∣ = d ∣ J 0 ∣ ≤ C t , q ( d − ∣ A ∣ ) |J_0|\le C_{t,q}(d-|A|) ∣ J 0 ∣ ≤ C t , q ( d − ∣ A ∣ )
the E E E ℓ s ∈ L S \ell_s\in\mathcal{L}_S ℓ s ∈ L S ∣ E ∩ [ ℓ s ] ∣ |E\cap[\ell_s]| ∣ E ∩ [ ℓ s ] ∣
for some global constant C t , q C_{t,q} C t , q
The key idea to proving the theorem is to observe that, even though the number of agents in A A A d d d u ∗ u_* u ∗ n n n i ∈ A i \in A i ∈ A p i p_i p i t t t
p i : = { P i ( π i , ℓ 1 ) , … , P i ( π i , ℓ t ) } , p_i := \{\mathcal{P}_i(\pi_i,\ell_1),\ldots,\mathcal{P}_i(\pi_i,\ell_t)\}, p i := { P i ( π i , ℓ 1 ) , … , P i ( π i , ℓ t )} , where { ℓ 1 , … , ℓ t } = L S \{\ell_1,\ldots,\ell_t\}=\mathcal{L}_S { ℓ 1 , … , ℓ t } = L S p ∗ , j p_{\ast,j} p ∗ , j j ∈ [ d ] j\in[d] j ∈ [ d ] u ∗ u_\ast u ∗
p ∗ , j : = { P u ∗ ( π u ∗ , j E , ℓ 1 ) , … , P u ∗ ( π u ∗ , j E , ℓ t ) } . p_{\ast,j} := \{\mathcal{P}_{u_\ast}(\pi^{E}_{u_\ast,j},\ell_1),\ldots,\mathcal{P}_{u_\ast}(\pi^{E}_{u_\ast,j},\ell_t)\}. p ∗ , j := { P u ∗ ( π u ∗ , j E , ℓ 1 ) , … , P u ∗ ( π u ∗ , j E , ℓ t )} . Note that both p i p_i p i p ∗ , j p_{\ast,j} p ∗ , j ℓ 1 < ⋯ < ℓ t \ell_1<\cdots<\ell_t ℓ 1 < ⋯ < ℓ t q q q q q q p i , p ∗ , j p_i,p_{\ast,j} p i , p ∗ , j
P = { p ∈ Z ≥ 0 t : 0 ≤ p 1 ≤ ⋯ ≤ p t ≤ q } . P=\{p\in\mathbb{Z}_{\geq0}^{t}:0\leq p_1\leq\cdots\leq p_t\leq q\}. P = { p ∈ Z ≥ 0 t : 0 ≤ p 1 ≤ ⋯ ≤ p t ≤ q } . We also define the load capacity vector as
E : = { ∣ E ∩ [ ℓ 1 ] ∣ , … , ∣ E ∩ [ ℓ t ] ∣ } . \mathcal{E}:=\{|E\cap[\ell_1]|,\ldots,|E\cap[\ell_t]|\}. E := { ∣ E ∩ [ ℓ 1 ] ∣ , … , ∣ E ∩ [ ℓ t ] ∣ } . We would like to find subsets A 0 ⊆ A A_0\subseteq A A 0 ⊆ A J 0 ⊆ [ d ] J_0\subseteq[d] J 0 ⊆ [ d ]
∣ A 0 ∣ + ∣ J 0 ∣ = d , and P ( A 0 , J 0 ) : = ∑ i ∈ A 0 p i + ∑ j ∈ J 0 p ∗ , j ≤ E , |A_0|+|J_0|=d,\qquad\text{and}\qquad\mathcal{P}(A_0,J_0):=\sum_{i\in A_0}p_i+\sum_{j\in J_0}p_{\ast,j}\leq\mathcal{E}, ∣ A 0 ∣ + ∣ J 0 ∣ = d , and P ( A 0 , J 0 ) := i ∈ A 0 ∑ p i + j ∈ J 0 ∑ p ∗ , j ≤ E , where the last inequality is component-wise.
We do this iteratively. At the beginning of each iteration k ≥ 1 k\geq1 k ≥ 1 A k ⊆ A A^k\subseteq A A k ⊆ A J k ⊆ [ d ] J^k\subseteq[d] J k ⊆ [ d ] P ( A k , J k ) ≤ E \mathcal{P}(A^k,J^k)\leq\mathcal{E} P ( A k , J k ) ≤ E A ′ ⊆ A k A'\subseteq A^k A ′ ⊆ A k A k A^k A k J ′ ⊆ [ d ] − J k J'\subseteq[d]-J^k J ′ ⊆ [ d ] − J k J k J^k J k ∣ J ′ ∣ = ∣ A ′ ∣ + 1 |J'|=|A'|+1 ∣ J ′ ∣ = ∣ A ′ ∣ + 1
P ( A k − A ′ , J k + J ′ ) ≤ E . \mathcal{P}(A^k-A',J^k+J')\leq\mathcal{E}. P ( A k − A ′ , J k + J ′ ) ≤ E . After this, we move to iteration k + 1 k+1 k + 1 A k + 1 = A k − A ′ A^{k+1}=A^k-A' A k + 1 = A k − A ′ J k + 1 = J k + J ′ J^{k+1}=J^k+J' J k + 1 = J k + J ′ A 1 = A A^1=A A 1 = A J 1 = ∅ J^1=\varnothing J 1 = ∅
P ( A , ∅ ) [ ℓ s ] = ∑ i ∈ A P i ( π i E , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣ = E [ ℓ s ] , ∀ ℓ s ∈ L S . \mathcal{P}(A,\varnothing)[\ell_s]=\sum_{i\in A}\mathcal{P}_i(\pi_i^E,\ell_s)\leq|E\cap[\ell_s]|=\mathcal{E}[\ell_s],\qquad\forall\ell_s\in\mathcal{L}_S. P ( A , ∅ ) [ ℓ s ] = i ∈ A ∑ P i ( π i E , ℓ s ) ≤ ∣ E ∩ [ ℓ s ] ∣ = E [ ℓ s ] , ∀ ℓ s ∈ L S . Suppose at iteration k k k ∣ A k ∣ + ∣ J k ∣ < d |A^k|+|J^k|<d ∣ A k ∣ + ∣ J k ∣ < d P ( A k , J k ) ≤ E \mathcal{P}(A^k,J^k)\leq\mathcal{E} P ( A k , J k ) ≤ E J ⊆ [ d ] − J k J\subseteq[d]-J^k J ⊆ [ d ] − J k ∣ J ∣ = ∣ A k ∣ + 1 |J|=|A^k|+1 ∣ J ∣ = ∣ A k ∣ + 1 d − ∣ J k ∣ ≥ ∣ A k ∣ + 1 d-|J^k|\geq|A^k|+1 d − ∣ J k ∣ ≥ ∣ A k ∣ + 1 J J J J k J^k J k A k A^k A k ( ∅ , J k + J ) (\varnothing,J^k+J) ( ∅ , J k + J )
P ( ∅ , J k + J ) = ∑ j ∈ J k + J p ∗ , j ≤ ∑ j = 1 d p ∗ , j = P ( ∅ , [ d ] ) ≤ E . \mathcal{P}(\varnothing,J^k+J)=\sum_{j\in J^k+J}p_{\ast,j}\leq\sum_{j=1}^{d}p_{\ast,j}=\mathcal{P}(\varnothing,[d])\leq\mathcal{E}. P ( ∅ , J k + J ) = j ∈ J k + J ∑ p ∗ , j ≤ j = 1 ∑ d p ∗ , j = P ( ∅ , [ d ]) ≤ E . The last inequality follows from the fact that all profiles { π u ∗ , j E : j ∈ [ d ] } \{\pi^E_{u_\ast,j}:j\in[d]\} { π u ∗ , j E : j ∈ [ d ]} { B u ∗ , j : j ∈ [ d ] } \{B_{u_\ast,j}:j\in[d]\} { B u ∗ , j : j ∈ [ d ]} E ∩ [ ℓ ] E\cap[\ell] E ∩ [ ℓ ] 0 ≤ ℓ ≤ m 0\leq\ell\leq m 0 ≤ ℓ ≤ m
The problem with replacement A k → ∅ A^k\to\varnothing A k → ∅ J k → J k + J J^k\to J^k+J J k → J k + J J J J d = Θ ( n ) d=\Theta(n) d = Θ ( n ) u ∗ u_\ast u ∗
Lemma C.1. For a global constant C ( t , q ) C(t,q) C ( t , q ) A ′ ⊆ A k A'\subseteq A^k A ′ ⊆ A k J ′ ⊆ J J'\subseteq J J ′ ⊆ J ∣ J ′ ∣ = ∣ A ′ ∣ + 1 |J'|=|A'|+1 ∣ J ′ ∣ = ∣ A ′ ∣ + 1 ∣ J ′ ∣ ≤ C t , q |J'|\leq C_{t,q} ∣ J ′ ∣ ≤ C t , q
P ( A k − A ′ , J k + J ′ ) ≤ max { P ( A k , J k ) , P ( ∅ , J k + J ) } ≤ E . \mathcal{P}(A^k-A',J^k+J')\leq\max\{\mathcal{P}(A^k,J^k),\mathcal{P}(\varnothing,J^k+J)\}\leq\mathcal{E}. P ( A k − A ′ , J k + J ′ ) ≤ max { P ( A k , J k ) , P ( ∅ , J k + J )} ≤ E . Proof. We will first define an abstract set of integral vectors, that depends only on values t , q t, q t , q
Let x , y ∈ Z ≥ 0 P x,y \in\mathbb{Z}_{\geq0}^{P} x , y ∈ Z ≥ 0 P p ∈ P p \in P p ∈ P x , y ∈ Z ≥ 0 P x,y \in\mathbb{Z}_{\geq0}^{P} x , y ∈ Z ≥ 0 P z + = z + ( x , y ) ∈ Z ≥ 0 t z_{+}=z_{+}(x,y)\in\mathbb{Z}_{\geq0}^{t} z + = z + ( x , y ) ∈ Z ≥ 0 t
z + : = ( ∑ p ∈ P p ⋅ y p − ∑ p ∈ P p ⋅ x p ) + . z_{+}:=\left(\sum_{p\in P}p\cdot y_{p}-\sum_{p\in P}p\cdot x_{p}\right)_{+}. z + := p ∈ P ∑ p ⋅ y p − p ∈ P ∑ p ⋅ x p + . Denote by R ( t , q ) \mathcal{R}(t,q) R ( t , q ) ( x , y , z + ) (x,y,z_{+}) ( x , y , z + )
∑ p ∈ P y p = ∑ p ∈ P x p + 1. \sum_{p\in P}y_{p}=\sum_{p\in P}x_{p}+1. p ∈ P ∑ y p = p ∈ P ∑ x p + 1. Observe that R ( t , q ) \mathcal{R}(t,q) R ( t , q ) Z ≥ 0 2 ∣ P ∣ + t \mathbb{Z}_{\geq0}^{2|P|+t} Z ≥ 0 2∣ P ∣ + t R ( t , q ) \mathcal{R}(t,q) R ( t , q ) t , q t,q t , q P = { p ∈ Z ≥ 0 t : 0 ≤ p 1 ≤ … ≤ p t ≤ q } P=\{p\in\mathbb{Z}_{\geq0}^{t}:0\leq p_{1}\leq\ldots\leq p_{t}\leq q\} P = { p ∈ Z ≥ 0 t : 0 ≤ p 1 ≤ … ≤ p t ≤ q }
While vectors x , y x,y x , y x x x A k A^{k} A k removing A ′ ⊆ A k A'\subseteq A^{k} A ′ ⊆ A k y y y J k J^{k} J k adding J ′ ⊆ J J'\subseteq J J ′ ⊆ J p ∈ P p\in P p ∈ P
x p x_{p} x p i ∈ A k i\in A^{k} i ∈ A k p p p A k A^{k} A k
y p y_{p} y p j ∈ J j\in J j ∈ J u ∗ u_{*} u ∗ p p p J k J^{k} J k
In this case, vector z + z_{+} z + ( A k , J k ) (A^{k},J^{k}) ( A k , J k )
Note that R ( t , q ) \mathcal{R}(t,q) R ( t , q ) ( x ∗ , y ∗ , z + ∗ ) (x^{*},y^{*},z^{*}_{+}) ( x ∗ , y ∗ , z + ∗ ) ( ∅ , J k + J ) (\varnothing,J^{k}+J) ( ∅ , J k + J ) p ∈ P p\in P p ∈ P
x p ∗ = ∣ { i ∈ A k : p i = p } ∣ , and y p ∗ = ∣ { j ∈ J : p ∗ , j = p } ∣ , x_{p}^{*}=|\{i\in A^{k}:p_{i}=p\}|,\qquad\text{and}\qquad y_{p}^{*}=|\{j\in J:p_{*,j}=p\}|, x p ∗ = ∣ { i ∈ A k : p i = p } ∣ , and y p ∗ = ∣ { j ∈ J : p ∗ , j = p } ∣ , and
z + ∗ = ( ∑ p ∈ P p ⋅ y p ∗ − ∑ p ∈ P p ⋅ x p ∗ ) + = ( ∑ j ∈ J p ∗ , j − ∑ i ∈ A k p i ) + = ( P ( ∅ , J ∪ J k ) − P ( A k , J k ) ) + . z_{+}^{*}=\left(\sum_{p\in P}p\cdot y_{p}^{*}-\sum_{p\in P}p\cdot x_{p}^{*}\right)_{+}
=\left(\sum_{j\in J}p_{*,j}-\sum_{i\in A^{k}}p_{i}\right)_{+}
=\left(\mathcal{P}(\varnothing,J\cup J^{k})-\mathcal{P}(A^{k},J^{k})\right)_{+}. z + ∗ = p ∈ P ∑ p ⋅ y p ∗ − p ∈ P ∑ p ⋅ x p ∗ + = j ∈ J ∑ p ∗ , j − i ∈ A k ∑ p i + = ( P ( ∅ , J ∪ J k ) − P ( A k , J k ) ) + . Since ∣ J ∣ = ∣ A k ∣ + 1 |J|=|A^{k}|+1 ∣ J ∣ = ∣ A k ∣ + 1 ∑ p ∈ P y p ∗ = ∑ p ∈ P x p ∗ + 1 \sum_{p\in P}y_{p}^{*}=\sum_{p\in P}x_{p}^{*}+1 ∑ p ∈ P y p ∗ = ∑ p ∈ P x p ∗ + 1 ( x ∗ , y ∗ , z + ∗ ) ∈ R ( t , q ) (x^{*},y^{*},z_{+}^{*})\in\mathcal{R}(t,q) ( x ∗ , y ∗ , z + ∗ ) ∈ R ( t , q )
We would like an element of R ( t , q ) \mathcal{R}(t,q) R ( t , q ) A k A^{k} A k J k J^{k} J k z z z E \mathcal{E} E J k J^{k} J k ∑ p ∈ P y p \sum_{p\in P}y_{p} ∑ p ∈ P y p
Lemma C.2 ([11 , 10 ] ). For K ≥ 1 K\geq1 K ≥ 1 Z ≥ 0 K \mathbb{Z}_{\geq0}^{K} Z ≥ 0 K
For completeness, a short proof of Lemma C.2 is given below (Lemma C.3).
Since R ( t , q ) \mathcal{R}(t,q) R ( t , q ) Z ≥ 0 ∣ P ∣ + t \mathbb{Z}_{\geq0}^{|P|+t} Z ≥ 0 ∣ P ∣ + t R ( t , q ) \mathcal{R}(t,q) R ( t , q ) { ( x r , y r , z + r ) } r = 1 R \{(x^{r},y^{r},z_{+}^{r})\}_{r=1}^{R} {( x r , y r , z + r ) } r = 1 R R ( t , q ) \mathcal{R}(t,q) R ( t , q ) t t t q q q R R R t t t q q q ( x r , y r , z + r ) (x^{r},y^{r},z_{+}^{r}) ( x r , y r , z + r ) r ∈ R r\in R r ∈ R t , q t,q t , q C t , q : = max r ∈ [ R ] ∑ p ∈ P y p r C_{t,q}:=\max_{r\in[R]}\sum_{p\in P}y_{p}^{r} C t , q := max r ∈ [ R ] ∑ p ∈ P y p r R ( t , q ) \mathcal{R}(t,q) R ( t , q )
Let ( x ∗ , y ∗ , z + ∗ ) ∈ R ( t , q ) (x^{*},y^{*},z_{+}^{*})\in\mathcal{R}(t,q) ( x ∗ , y ∗ , z + ∗ ) ∈ R ( t , q ) A k A^{k} A k J J J ( x r , y r , z + r ) ∈ R ( t , q ) (x^{r},y^{r},z_{+}^{r})\in\mathcal{R}(t,q) ( x r , y r , z + r ) ∈ R ( t , q ) ( x r , y r , z + r ) ≤ ( x ∗ , y ∗ , z + ∗ ) (x^{r},y^{r},z_{+}^{r})\leq(x^{*},y^{*},z_{+}^{*}) ( x r , y r , z + r ) ≤ ( x ∗ , y ∗ , z + ∗ )
∑ p ∈ P y p r ≤ C t , q . \sum_{p\in P}y_{p}^{r}\leq C_{t,q}. p ∈ P ∑ y p r ≤ C t , q . Since x r ≤ x ∗ x^{r} \leq x^{*} x r ≤ x ∗ p ∈ P p \in P p ∈ P x p r ≤ x p ∗ x^{r}_{p} \leq x^{*}_{p} x p r ≤ x p ∗ x r x^{r} x r p p p A k A^{k} A k p p p A k A^{k} A k A ′ ⊆ A k A' \subseteq A^{k} A ′ ⊆ A k x p r x^{r}_{p} x p r i ∈ A k i \in A^{k} i ∈ A k p p p p ∈ P p \in P p ∈ P y r ≤ y ∗ y^{r} \leq y^{*} y r ≤ y ∗ J ′ ⊆ J J' \subseteq J J ′ ⊆ J y p r y^{r}_{p} y p r j ∈ J j \in J j ∈ J p p p p ∈ P p \in P p ∈ P ∣ J ′ ∣ = ∣ A ′ ∣ + 1 |J'| = |A'| + 1 ∣ J ′ ∣ = ∣ A ′ ∣ + 1 ∣ J ′ ∣ = ∑ p ∈ P y p r ≤ C ( t , q ) |J'| = \sum_{p \in P} y^{r}_{p} \leq C(t,q) ∣ J ′ ∣ = ∑ p ∈ P y p r ≤ C ( t , q ) z + r ≤ z + ∗ z^{r}_{+} \leq z^{*}_{+} z + r ≤ z + ∗
( ∑ j ∈ J ′ p ∗ , j − ∑ i ∈ A ′ p i ) + ≤ ( P ( ∅ , J k + J ) − P ( A k , J k ) ) + . \left(\sum_{j\in J'}p_{*,j}-\sum_{i\in A'}p_i\right)_{+}\leq\left(\mathcal{P}(\varnothing,J^k+J)-\mathcal{P}(A^k,J^k)\right)_{+}. j ∈ J ′ ∑ p ∗ , j − i ∈ A ′ ∑ p i + ≤ ( P ( ∅ , J k + J ) − P ( A k , J k ) ) + . It follows that
P ( A k − A ′ , J k + J ′ ) ≤ P ( A k , J k ) + ( ∑ j ∈ J ′ p ∗ , j − ∑ i ∈ A ′ p i ) + = P ( A k , J k ) + ( P ( ∅ , J k + J ) − P ( A k , J k ) ) + = max { P ( A k , J k ) , P ( ∅ , J k + J ) } ≤ E . \begin{aligned}
\mathcal{P}(A^k-A',J^k+J')&\leq\mathcal{P}(A^k,J^k)+\left(\sum_{j\in J'}p_{*,j}-\sum_{i\in A'}p_i\right)_{+}\\
&=\mathcal{P}(A^k,J^k)+\left(\mathcal{P}(\varnothing,J^k+J)-\mathcal{P}(A^k,J^k)\right)_{+}\\
&=\max\{\mathcal{P}(A^k,J^k),\mathcal{P}(\varnothing,J^k+J)\}\leq\mathcal{E}.
\end{aligned} P ( A k − A ′ , J k + J ′ ) ≤ P ( A k , J k ) + j ∈ J ′ ∑ p ∗ , j − i ∈ A ′ ∑ p i + = P ( A k , J k ) + ( P ( ∅ , J k + J ) − P ( A k , J k ) ) + = max { P ( A k , J k ) , P ( ∅ , J k + J )} ≤ E . At iteration k ≥ 1 k \geq1 k ≥ 1 ( A k , J k ) (A^k,J^k) ( A k , J k ) A ′ ⊆ A k A' \subseteq A^k A ′ ⊆ A k J ′ ⊆ J ⊆ [ d ] − J k J' \subseteq J \subseteq[d]-J^k J ′ ⊆ J ⊆ [ d ] − J k ∣ J ′ ∣ = ∣ A ′ ∣ + 1 |J'| = |A'|+1 ∣ J ′ ∣ = ∣ A ′ ∣ + 1 ∣ J ′ ∣ ≤ C t , q |J'| \leq C_{t,q} ∣ J ′ ∣ ≤ C t , q P ( A k − A ′ , J k + J ′ ) ≤ E \mathcal{P}(A^k-A',J^k+J') \leq\mathcal{E} P ( A k − A ′ , J k + J ′ ) ≤ E A k + 1 = A k − A ′ A^{k+1}=A^k-A' A k + 1 = A k − A ′ J k + 1 = J k + J ′ J^{k+1}=J^k+J' J k + 1 = J k + J ′ C t , q C_{t,q} C t , q u ∗ u_{*} u ∗ d − ∣ A ∣ d-|A| d − ∣ A ∣ A 0 : = A d − ∣ A ∣ + 1 A_0:=A^{d-|A|+1} A 0 := A d − ∣ A ∣ + 1 J 0 : = J d − ∣ A ∣ + 1 J_0:=J^{d-|A|+1} J 0 := J d − ∣ A ∣ + 1 P ( A 0 , J 0 ) ≤ E \mathcal{P}(A_0,J_0)\leq\mathcal{E} P ( A 0 , J 0 ) ≤ E ∣ A 0 ∣ + ∣ J 0 ∣ = d |A_0|+|J_0|=d ∣ A 0 ∣ + ∣ J 0 ∣ = d ∣ J 0 ∣ ≤ C t , q ( d − ∣ A ∣ ) |J_0|\leq C_{t,q}(d-|A|) ∣ J 0 ∣ ≤ C t , q ( d − ∣ A ∣ )
Lemma C.3 (Short proof of Dickson’s lemma, [11 , 10 ] ). For K ≥ 1 K \geq1 K ≥ 1 Z ≥ 0 K \mathbb{Z}_{\geq0}^{K} Z ≥ 0 K
Proof. Let S ⊆ Z ≥ 0 K S \subseteq\mathbb{Z}_{\geq0}^{K} S ⊆ Z ≥ 0 K y ∈ S y \in S y ∈ S x ∈ S − { y } x \in S-\{y\} x ∈ S − { y } x ≤ y x \leq y x ≤ y
It is well-known that every infinite sequence of non-negative integers has an infinite non-decreasing subsequence. Then, every infinite sequence x 1 , x 2 , … ∈ Z ≥ 0 K x^1,x^2,\ldots\in\mathbb{Z}_{\geq0}^{K} x 1 , x 2 , … ∈ Z ≥ 0 K K K K x i 1 ≤ x i 2 ≤ … ∈ Z ≥ 0 K x^{i_1}\leq x^{i_2}\leq\ldots\in\mathbb{Z}_{\geq0}^{K} x i 1 ≤ x i 2 ≤ … ∈ Z ≥ 0 K
Suppose that S S S x , y ∈ S x,y\in S x , y ∈ S x ≤ y x\leq y x ≤ y S S S
Next, fix x ∈ S x\in S x ∈ S S x : = { y ∈ S : y ≤ x } S_x:=\{y\in S:y\leq x\} S x := { y ∈ S : y ≤ x } x ∈ S x x\in S_x x ∈ S x y ∈ S x y\in S_x y ∈ S x x x x K K K y ∈ S x y\in S_x y ∈ S x ∑ k = 1 K y k \sum_{k=1}^{K}y_k ∑ k = 1 K y k z ∈ S − { y } z\in S-\{y\} z ∈ S − { y } z ≤ y z\leq y z ≤ y z ≤ x z\leq x z ≤ x z ∈ S x z\in S_x z ∈ S x z z z y y y ∑ k = 1 K z k ≤ ∑ k = 1 K y k \sum_{k=1}^{K}z_k\leq\sum_{k=1}^{K}y_k ∑ k = 1 K z k ≤ ∑ k = 1 K y k y y y S S S y ≤ x y\leq x y ≤ x
Constant estimate
Theorem D.1. For t ≥ 1 t \geq1 t ≥ 1 q ≥ 2 q \geq2 q ≥ 2 C t , q C_{t,q} C t , q
C t , q = ( t + q q ) ( t + 1 ) q t . C_{t,q}=\binom{t+q}{q}(t+1)q^{t}. C t , q = ( q t + q ) ( t + 1 ) q t . Proof. Let P = { p ∈ Z ≥ 0 t : 0 ≤ p 1 ≤ … ≤ p t ≤ q } P = \{p \in\mathbb{Z}_{\geq0}^{t} : 0 \leq p_1 \leq\ldots\leq p_t \leq q\} P = { p ∈ Z ≥ 0 t : 0 ≤ p 1 ≤ … ≤ p t ≤ q } L S = { ℓ 1 , … , ℓ t } \mathcal{L}_S = \{\ell_1, \ldots, \ell_t\} L S = { ℓ 1 , … , ℓ t } t t t
∣ P ∣ = ( t + q q ) . |P| = \binom{t+q}{q}. ∣ P ∣ = ( q t + q ) . Recall from Lemma C.1 that set R ( x , y ) \mathcal{R}(x,y) R ( x , y ) ( x , y , z + ) (x,y,z_+) ( x , y , z + )
∑ p ∈ P y p = ∑ p ∈ P x p + 1 , z + = ( ∑ p ∈ P p ⋅ y p − ∑ p ∈ P p ⋅ x p ) + . \sum_{p\in P} y_p = \sum_{p\in P} x_p + 1, \qquad z_+ = \left(\sum_{p\in P} p\cdot y_p - \sum_{p\in P} p\cdot x_p\right)_+. p ∈ P ∑ y p = p ∈ P ∑ x p + 1 , z + = p ∈ P ∑ p ⋅ y p − p ∈ P ∑ p ⋅ x p + . It suffices to show that every minimal element ( x r , y r , z + r ) (x^r,y^r,z^r_+) ( x r , y r , z + r ) R ( t , q ) \mathcal{R}(t,q) R ( t , q )
∑ p ∈ P y p r ≤ ( t + q q ) ( t + 1 ) q t . \sum_{p\in P} y^r_p \leq\binom{t+q}{q}(t+1)q^t. p ∈ P ∑ y p r ≤ ( q t + q ) ( t + 1 ) q t . To prove this bound, we will need the following result on norm of indecomposable kernel vectors.
Theorem D.2 ([23 ] , Lemma 3.20). Let M M M k k k t t t
Δ ( M ) : = max { ∣ det M ′ ∣ : M ′ is a square submatrix of M } . \Delta(M) := \max\{|\det M'| : M' \text{ is a square submatrix of } M\}. Δ ( M ) := max { ∣ det M ′ ∣ : M ′ is a square submatrix of M } . Suppose some v ∈ Z ≥ 0 k v \in\mathbb{Z}_{\geq0}^{k} v ∈ Z ≥ 0 k v ≠ 0 v \ne0 v = 0 M v = 0 Mv = 0 M v = 0 u ∈ Z ≥ 0 k u \in\mathbb{Z}_{\geq0}^{k} u ∈ Z ≥ 0 k 0 ≤ u ≤ v 0 \leq u \leq v 0 ≤ u ≤ v M u = 0 Mu = 0 M u = 0 u = 0 u = 0 u = 0 u = v u = v u = v
∥ v ∥ 1 ≤ ( k − t ) ( t + 1 ) Δ ( M ) . \lVert v\rVert_1 \leq(k-t)(t+1)\Delta(M). ∥ v ∥ 1 ≤ ( k − t ) ( t + 1 ) Δ ( M ) . We will define a certain matrix M M M k = ∣ P ∣ + t k = |P|+t k = ∣ P ∣ + t t t t Δ ( M ) ≤ q t \Delta(M) \leq q^t Δ ( M ) ≤ q t R ( t , q ) \mathcal{R}(t,q) R ( t , q ) M M M Theorem D.2 .
Fix the minimal element ( x r , y r , z + r ) ∈ R ( t , q ) (x^r,y^r,z^r_+) \in\mathcal{R}(t,q) ( x r , y r , z + r ) ∈ R ( t , q ) p ∈ P p \in P p ∈ P x p r y p r = 0 x^r_p y^r_p = 0 x p r y p r = 0 R ( t , q ) \mathcal{R}(t,q) R ( t , q ) z r = ∑ p ∈ P p ⋅ y p r − ∑ p ∈ P p ⋅ x p r z^r = \sum_{p\in P} p\cdot y^r_p - \sum_{p\in P} p\cdot x^r_p z r = ∑ p ∈ P p ⋅ y p r − ∑ p ∈ P p ⋅ x p r ( z r ) + = z + r (z^r)_+ = z^r_+ ( z r ) + = z + r p ∈ P p \in P p ∈ P
a p : = { − p , x p r > 0 ; p , x p r = 0. a_p := \begin{cases}
-p, & x^r_p > 0;\\
p, & x^r_p = 0.
\end{cases} a p := { − p , p , x p r > 0 ; x p r = 0. Note that ( x p r + y p r ) ⋅ a p = ( y p r − x p r ) ⋅ p (x^r_p+y^r_p)\cdot a_p = (y^r_p-x^r_p)\cdot p ( x p r + y p r ) ⋅ a p = ( y p r − x p r ) ⋅ p s ∈ [ t ] s \in[t] s ∈ [ t ] e s e_s e s s s s
b s : = { − e s , z s r > 0 ; e s , z s r ≤ 0. b_s := \begin{cases}
-e_s, & z^r_s > 0;\\
e_s, & z^r_s \leq0.
\end{cases} b s := { − e s , e s , z s r > 0 ; z s r ≤ 0. This way, ∑ s = 1 t b s ⋅ ∣ z s r ∣ = − z r \sum_{s=1}^{t} b_s\cdot|z^r_s| = -z^r ∑ s = 1 t b s ⋅ ∣ z s r ∣ = − z r ∣ P ∣ + t |P|+t ∣ P ∣ + t M 0 M_0 M 0 v v v
M 0 = [ { a p } p ∈ P , b 1 , … , b t ] , v = [ { x p r + y p r } p ∈ P , ∣ z 1 r ∣ , … , ∣ z t r ∣ ] . M_0 = [\{a_p\}_{p\in P},b_1,\ldots,b_t], \qquad v = [\{x^r_p+y^r_p\}_{p\in P}, |z^r_1|,\ldots,|z^r_t|]. M 0 = [{ a p } p ∈ P , b 1 , … , b t ] , v = [{ x p r + y p r } p ∈ P , ∣ z 1 r ∣ , … , ∣ z t r ∣ ] . The vector v v v M 0 v = 0 M_0v = 0 M 0 v = 0 M 0 M_0 M 0 t t t
Lemma D.1. Vector v v v M 0 M_0 M 0 Theorem D.2 ).
Proof. Suppose the contrary, i.e that v = v ′ + v ′ ′ v = v' + v'' v = v ′ + v ′′ v ′ , v ′ ′ ≠ 0 v',v'' \ne0 v ′ , v ′′ = 0 M 0 v ′ = 0 = M 0 v ′ ′ M_0v' = 0 = M_0v'' M 0 v ′ = 0 = M 0 v ′′ ( x r , y r , z + r ) (x^r,y^r,z^r_+) ( x r , y r , z + r ) ( x ^ , y ^ , z ^ + ) ∈ R ( t , q ) (\hat{x},\hat{y},\hat{z}_+) \in\mathcal{R}(t,q) ( x ^ , y ^ , z ^ + ) ∈ R ( t , q ) ( x r , y r , z + r ) (x^r,y^r,z^r_+) ( x r , y r , z + r ) p ∈ P p \in P p ∈ P x ′ , x ′ ′ x^\prime,x^{\prime\prime} x ′ , x ′′ y ′ , y ′ ′ y^\prime,y^{\prime\prime} y ′ , y ′′
x p ′ = v p ′ 1 { x p r > 0 } , x p ′ ′ = v p ′ ′ 1 { x p r > 0 } , y p ′ = v p ′ 1 { y p r > 0 } , y p ′ ′ = v p ′ ′ 1 { y p r > 0 } . x^\prime_p=v^\prime_p\mathbf{1}\{x^r_p>0\},\qquad x^{\prime\prime}_p=v^{\prime\prime}_p\mathbf{1}\{x^r_p>0\},\qquad y^\prime_p=v^\prime_p\mathbf{1}\{y^r_p>0\},\qquad y^{\prime\prime}_p=v^{\prime\prime}_p\mathbf{1}\{y^r_p>0\}. x p ′ = v p ′ 1 { x p r > 0 } , x p ′′ = v p ′′ 1 { x p r > 0 } , y p ′ = v p ′ 1 { y p r > 0 } , y p ′′ = v p ′′ 1 { y p r > 0 } . It is clear that x ′ + x ′ ′ = x r x^\prime+x^{\prime\prime}=x^r x ′ + x ′′ = x r y ′ + y ′ ′ = y r y^\prime+y^{\prime\prime}=y^r y ′ + y ′′ = y r z ′ , z ′ ′ ∈ Z t z^\prime,z^{\prime\prime}\in\mathbb{Z}^t z ′ , z ′′ ∈ Z t
z ′ = ∑ p ∈ P p ⋅ y p ′ − ∑ p ∈ P p ⋅ x p ′ , z ′ ′ = ∑ p ∈ P p ⋅ y p ′ ′ − ∑ p ∈ P p ⋅ x p ′ ′ . z^\prime=\sum_{p\in P}p\cdot y^\prime_p-\sum_{p\in P}p\cdot x^\prime_p,\qquad z^{\prime\prime}=\sum_{p\in P}p\cdot y^{\prime\prime}_p-\sum_{p\in P}p\cdot x^{\prime\prime}_p. z ′ = p ∈ P ∑ p ⋅ y p ′ − p ∈ P ∑ p ⋅ x p ′ , z ′′ = p ∈ P ∑ p ⋅ y p ′′ − p ∈ P ∑ p ⋅ x p ′′ . Since M 0 v ′ = 0 = M 0 v ′ ′ M_0v^\prime=0=M_0v^{\prime\prime} M 0 v ′ = 0 = M 0 v ′′ b s b_s b s s ∈ [ t ] s\in[t] s ∈ [ t ]
z s ′ = { v s ′ , z s r > 0 ; − v s ′ , z s r ≤ 0 , z s ′ ′ = { v s ′ ′ , z s r > 0 ; − v s ′ ′ , z s r ≤ 0. z^\prime_s=
\begin{cases}
v^\prime_s,&z^r_s>0;\\
-v^\prime_s,&z^r_s\leq0,
\end{cases}
\qquad
z^{\prime\prime}_s=
\begin{cases}
v^{\prime\prime}_s,&z^r_s>0;\\
-v^{\prime\prime}_s,&z^r_s\leq0.
\end{cases} z s ′ = { v s ′ , − v s ′ , z s r > 0 ; z s r ≤ 0 , z s ′′ = { v s ′′ , − v s ′′ , z s r > 0 ; z s r ≤ 0. Since v s ′ ≤ v s = ∣ z s r ∣ v^\prime_s\leq v_s=|z^r_s| v s ′ ≤ v s = ∣ z s r ∣ v s ′ ′ ≤ v s = ∣ z s r ∣ v^{\prime\prime}_s\leq v_s=|z^r_s| v s ′′ ≤ v s = ∣ z s r ∣ ( z ′ ) + ≤ ( z r ) + (z^\prime)_+\leq(z^r)_+ ( z ′ ) + ≤ ( z r ) + ( z ′ ′ ) + ≤ ( z r ) + (z^{\prime\prime})_+\leq(z^r)_+ ( z ′′ ) + ≤ ( z r ) +
Next, we claim that v ′ v^\prime v ′ v p ′ ≥ 1 v^\prime_p\geq1 v p ′ ≥ 1 p ∈ P p\in P p ∈ P v ′ = [ 0 , v 1 ′ , … , v t ′ ] v^\prime=[0,v^\prime_1,\ldots,v^\prime_t] v ′ = [ 0 , v 1 ′ , … , v t ′ ] 0 = M 0 v ′ = ∑ s = 1 t v s ′ ⋅ b s 0=M_0v^\prime=\sum_{s=1}^t v^\prime_s\cdot b_s 0 = M 0 v ′ = ∑ s = 1 t v s ′ ⋅ b s b s b_s b s ± e s \pm e_s ± e s v s ′ = 0 v^\prime_s=0 v s ′ = 0 s ∈ [ t ] s\in[t] s ∈ [ t ] v ′ = 0 v^\prime=0 v ′ = 0 v ′ ′ v^{\prime\prime} v ′′ v p ′ ′ ≥ 1 v^{\prime\prime}_p\geq1 v p ′′ ≥ 1 p ∈ P p\in P p ∈ P
∑ p ∈ P ( x p ′ + y p ′ ) ≥ 1 , ∑ p ∈ P ( x p ′ ′ + y p ′ ′ ) ≥ 1. \sum_{p\in P}(x^\prime_p+y^\prime_p)\geq1,\qquad\sum_{p\in P}(x^{\prime\prime}_p+y^{\prime\prime}_p)\geq1. p ∈ P ∑ ( x p ′ + y p ′ ) ≥ 1 , p ∈ P ∑ ( x p ′′ + y p ′′ ) ≥ 1. Define (possibly negative) integers
g ′ : = ∑ p ∈ P y p ′ − ∑ p ∈ P x p ′ , g ′ ′ : = ∑ p ∈ P y p ′ ′ − ∑ p ∈ P x p ′ ′ . g^\prime:=\sum_{p\in P}y^\prime_p-\sum_{p\in P}x^\prime_p,\qquad g^{\prime\prime}:=\sum_{p\in P}y^{\prime\prime}_p-\sum_{p\in P}x^{\prime\prime}_p. g ′ := p ∈ P ∑ y p ′ − p ∈ P ∑ x p ′ , g ′′ := p ∈ P ∑ y p ′′ − p ∈ P ∑ x p ′′ . Since x ′ + x ′ ′ = x x^\prime+x^{\prime\prime}=x x ′ + x ′′ = x y ′ + y ′ ′ = y y^\prime+y^{\prime\prime}=y y ′ + y ′′ = y ∑ p ∈ P y p = ∑ p ∈ P x p + 1 \sum_{p\in P}y_p=\sum_{p\in P}x_p+1 ∑ p ∈ P y p = ∑ p ∈ P x p + 1 g ′ + g ′ ′ = 1 g^\prime+g^{\prime\prime}=1 g ′ + g ′′ = 1
Without loss of generality, assume g ′ ≥ 1 g^\prime\geq1 g ′ ≥ 1 g ′ ′ = 1 − g ′ ≤ 0 g^{\prime\prime}=1-g^\prime\leq0 g ′′ = 1 − g ′ ≤ 0
∑ p ∈ P y p ′ ′ ≤ ∑ p ∈ P x p ′ ′ . \sum_{p\in P}y^{\prime\prime}_p\leq\sum_{p\in P}x^{\prime\prime}_p. p ∈ P ∑ y p ′′ ≤ p ∈ P ∑ x p ′′ . Since ∑ p ∈ P ( x p ′ ′ + y p ′ ′ ) ≥ 1 \sum_{p\in P}(x^{\prime\prime}_p+y^{\prime\prime}_p)\geq1 ∑ p ∈ P ( x p ′′ + y p ′′ ) ≥ 1 ∑ p ∈ P x p ′ ′ ≥ 1 \sum_{p\in P}x^{\prime\prime}_p\geq1 ∑ p ∈ P x p ′′ ≥ 1 p ∈ P p\in P p ∈ P x p ′ ′ > 0 x^{\prime\prime}_p>0 x p ′′ > 0 x p ′ = x p r − x p ′ ′ < x p r x^\prime_p=x^r_p-x^{\prime\prime}_p<x^r_p x p ′ = x p r − x p ′′ < x p r x ′ ≤ x r x^\prime\leq x^r x ′ ≤ x r p ∈ P p\in P p ∈ P
It remains to create an element of R ( t , q ) \mathcal{R}(t,q) R ( t , q ) x ′ x^\prime x ′ ( x r , y r , z + r ) (x^r,y^r,z^r_+) ( x r , y r , z + r ) g ′ − 1 g^\prime-1 g ′ − 1 y ′ y^\prime y ′ ∑ p ∈ P y p ′ = ∑ p ∈ P x p ′ + g ′ \sum_{p\in P}y^\prime_p=\sum_{p\in P}x^\prime_p+g^\prime ∑ p ∈ P y p ′ = ∑ p ∈ P x p ′ + g ′ g ′ ≥ 1 g^\prime\geq1 g ′ ≥ 1 y ^ ≤ y ′ \hat{y}\leq y^\prime y ^ ≤ y ′
∑ p ∈ P y ^ p = ∑ p ∈ P x p ′ + 1. \sum_{p\in P}\hat{y}_p=\sum_{p\in P}x^\prime_p+1. p ∈ P ∑ y ^ p = p ∈ P ∑ x p ′ + 1. Since removing y y y z z z
z ^ : = ∑ p ∈ P p ⋅ y ^ p − ∑ p ∈ P p ⋅ x p ′ ≤ z ′ , \hat{z}:=\sum_{p\in P}p\cdot\hat{y}_p-\sum_{p\in P}p\cdot x^\prime_p\leq z^\prime, z ^ := p ∈ P ∑ p ⋅ y ^ p − p ∈ P ∑ p ⋅ x p ′ ≤ z ′ , It follows that ( z ^ ) + ≤ ( z ′ ) + ≤ ( z r ) + = z + r (\hat{z})_+\leq(z^\prime)_+\leq(z^r)_+=z^r_+ ( z ^ ) + ≤ ( z ′ ) + ≤ ( z r ) + = z + r ( x ′ , y ^ , z ^ + ) ∈ R ( t , q ) (x^\prime,\hat{y},\hat{z}_+)\in\mathcal{R}(t,q) ( x ′ , y ^ , z ^ + ) ∈ R ( t , q ) ( x ′ , y ^ , z ^ + ) ≤ ( x r , y r , z + r ) (x^\prime,\hat{y},\hat{z}_+)\leq(x^r,y^r,z^r_+) ( x ′ , y ^ , z ^ + ) ≤ ( x r , y r , z + r ) p ∈ P p\in P p ∈ P x p ′ < x p r x^\prime_p<x^r_p x p ′ < x p r ( x ′ , y ^ , z ^ + ) (x^\prime,\hat{y},\hat{z}_+) ( x ′ , y ^ , z ^ + ) ( x r , y r , z + r ) (x^r,y^r,z^r_+) ( x r , y r , z + r ) R ( t , q ) \mathcal{R}(t,q) R ( t , q ) ( x r , y r , z + r ) (x^r,y^r,z^r_+) ( x r , y r , z + r )
We will now deal with matrix M 0 M_0 M 0 M 0 M_0 M 0 M M M L L L u u u
L u = [ u 1 , u 2 − u 1 , … , u t − u t − 1 ] , Lu=[u_1,u_2-u_1,\ldots,u_t-u_{t-1}], Lu = [ u 1 , u 2 − u 1 , … , u t − u t − 1 ] , and set M : = L M 0 M := LM_0 M := L M 0 M M M t t t ker M = ker M 0 \ker M = \ker M_0 ker M = ker M 0 v v v M M M p ∈ P p \in P p ∈ P L p = ( p 1 , p 2 − p 1 , … , p t − p t − 1 ) Lp = (p_1, p_2 - p_1, \ldots, p_t - p_{t-1}) L p = ( p 1 , p 2 − p 1 , … , p t − p t − 1 ) p t ≤ q p_t \le q p t ≤ q p ∈ P p \in P p ∈ P
∥ L ⋅ a p ∥ 2 = ∥ L ⋅ p ∥ 2 ≤ ∥ L ⋅ p ∥ 1 = p t ≤ q . \lVert L \cdot a_p\rVert_2 = \lVert L \cdot p\rVert_2 \le\lVert L \cdot p\rVert_1 = p_t \le q. ∥ L ⋅ a p ∥ 2 = ∥ L ⋅ p ∥ 2 ≤ ∥ L ⋅ p ∥ 1 = p t ≤ q . For the remaining columns, for every s ∈ [ t ] s \in[t] s ∈ [ t ]
L e s = { e s − e s + 1 , s < t , e t , s = t . Le_s =
\begin{cases}
e_s-e_{s+1}, & s<t,\\
e_t, & s=t.
\end{cases} L e s = { e s − e s + 1 , e t , s < t , s = t . So, L b s = ± L e s Lb_s = \pm Le_s L b s = ± L e s ∥ L b s ∥ 2 ≤ 2 ≤ q \lVert Lb_s\rVert_2 \le\sqrt{2} \le q ∥ L b s ∥ 2 ≤ 2 ≤ q M M M q q q Theorem D.2 , we have
∥ v ∥ 1 ≤ ( ( ∣ P ∣ + t ) − t ) ( t + 1 ) Δ ( M ) = ∣ P ∣ ( t + 1 ) Δ ( M ) . \lVert v\rVert_1 \le((|P|+t)-t)(t+1)\Delta(M)=|P|(t+1)\Delta(M). ∥ v ∥ 1 ≤ (( ∣ P ∣ + t ) − t ) ( t + 1 ) Δ ( M ) = ∣ P ∣ ( t + 1 ) Δ ( M ) . By definition, Δ ( M ) \Delta(M) Δ ( M ) M M M A A A k k k ( A 1 , … , A k ) (A^1,\ldots,A^k) ( A 1 , … , A k )
∣ det A ∣ ≤ ∏ j = 1 k ∥ A j ∥ 2 . |\det A| \le\prod_{j=1}^{k}\lVert A^j\rVert_2. ∣ det A ∣ ≤ j = 1 ∏ k ∥ A j ∥ 2 . Let M ′ M' M ′ M M M M ′ M' M ′ t t t det M ′ = 0 \det M'=0 det M ′ = 0 M M M t t t M ′ M' M ′ t t t ∣ det M ′ ∣ ≤ q t |\det M'| \le q^t ∣ det M ′ ∣ ≤ q t
∥ v ∥ 1 ≤ ∣ P ∣ ( t + 1 ) Δ ( M ) ≤ ( t + q q ) ( t + 1 ) q t . \lVert v\rVert_1 \le|P|(t+1)\Delta(M) \le\binom{t+q}{q}(t+1)q^t. ∥ v ∥ 1 ≤ ∣ P ∣ ( t + 1 ) Δ ( M ) ≤ ( q t + q ) ( t + 1 ) q t . Finally, by definition of v v v ∑ p ∈ P y p r ≤ ∥ v ∥ 1 \sum_{p\in P}y_p^r \le\lVert v\rVert_1 ∑ p ∈ P y p r ≤ ∥ v ∥ 1
Corollary D.1. For every ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 ) q ≥ 3 q \ge3 q ≥ 3 n ε , q n_{\varepsilon,q} n ε , q
n ε , q = ⌈ 2 64 K ⌉ for K = q 6 log q ε . n_{\varepsilon,q}=\left\lceil2^{64K}\right\rceil
\qquad\text{for}\qquad
K=\frac{q^6\log q}{\varepsilon}. n ε , q = ⌈ 2 64 K ⌉ for K = ε q 6 log q . Proof. The estimates will be crude. Recall our parameters γ = min ( ε , 1 / ( 2 q ) ) \gamma=\min(\varepsilon,1/(2q)) γ = min ( ε , 1/ ( 2 q )) λ = 2 q 2 q − 1 \lambda=\frac{2q}{2q-1} λ = 2 q − 1 2 q
ρ q , n : = 8 + 4 log λ ( q 3 d 2 ( d + 2 ) n ) + 8 q log ( ( d + 2 ) n ) . \rho_{q,n}:=8+4\log_{\lambda}\left(q^3d^2(d+2)n\right)+8q\log\left((d+2)n\right). ρ q , n := 8 + 4 log λ ( q 3 d 2 ( d + 2 ) n ) + 8 q log ( ( d + 2 ) n ) . Assume for now that n ≥ max { q , 16 / ε } n\ge\max\{q,16/\varepsilon\} n ≥ max { q , 16/ ε } ∣ L S ∣ |\mathcal{L}_S| ∣ L S ∣ t ≤ 2 q 2 t\le2q^2 t ≤ 2 q 2 C q C_q C q
C q = ( 2 q 2 + q q ) ( 2 q 2 + 1 ) q 2 q 2 . C_q=\binom{2q^2+q}{q}(2q^2+1)q^{2q^2}. C q = ( q 2 q 2 + q ) ( 2 q 2 + 1 ) q 2 q 2 . For q ≥ 3 q\ge3 q ≥ 3 ( 2 q 2 + q q ) ≤ q 3 q \binom{2q^2+q}{q}\le q^{3q} ( q 2 q 2 + q ) ≤ q 3 q 2 q 2 + 1 ≤ q 3 2q^2+1\le q^3 2 q 2 + 1 ≤ q 3 C 1 C_1 C 1
C q ≤ q 4 q 2 and C 1 = 1 + ⌈ 12 q 3 C q ⌉ ≤ q 5 q 2 . C_q\le q^{4q^2}
\qquad\text{and}\qquad
C_1=1+\lceil12q^3C_q\rceil\le q^{5q^2}. C q ≤ q 4 q 2 and C 1 = 1 + ⌈ 12 q 3 C q ⌉ ≤ q 5 q 2 . Since d ≤ n d\le n d ≤ n q 3 d 2 ( d + 2 ) n ≤ 2 q 3 n 4 q^3d^2(d+2)n\le2q^3n^4 q 3 d 2 ( d + 2 ) n ≤ 2 q 3 n 4 ( d + 2 ) n ≤ 2 n 2 (d+2)n\le2n^2 ( d + 2 ) n ≤ 2 n 2 log λ ≥ 1 / ( 2 q ) \log\lambda\ge1/(2q) log λ ≥ 1/ ( 2 q )
ρ q , n ≤ 9 + 8 q log ( 2 q 3 n 4 ) + 8 q ( 2 log n + 2 ) ≤ 80 q log n . \rho_{q,n}\le9+8q\log(2q^3n^4)+8q(2\log n+2)\le80q\log n. ρ q , n ≤ 9 + 8 q log ( 2 q 3 n 4 ) + 8 q ( 2 log n + 2 ) ≤ 80 q log n . Therefore, by the choice of r 1 r_1 r 1
r 1 : = 1 + ⌈ ( 2 q ( q − 1 ) γ + 2 ) ρ q , n ⌉ ≤ 256 q 2 ( q − 1 ) γ log n ≤ 512 q 4 ε log n . r_1 := 1 + \left\lceil\left(\frac{2q(q-1)}{\gamma}+2\right)\rho_{q,n}\right\rceil\le\frac{256q^2(q-1)}{\gamma}\log n \le\frac{512q^4}{\varepsilon}\log n. r 1 := 1 + ⌈ ( γ 2 q ( q − 1 ) + 2 ) ρ q , n ⌉ ≤ γ 256 q 2 ( q − 1 ) log n ≤ ε 512 q 4 log n . By Lemma 4.1 we have the bound on ∣ S ∣ |S| ∣ S ∣
∣ S ∣ ≤ 2 q 2 C 1 8 q 2 / ε r 1 ≤ 1024 q 6 ε ⋅ q 4 q 4 / ε log n ≤ q 42 q 4 / ε log n . |S| \le2q^2 C_1^{8q^2/\varepsilon}r_1 \le\frac{1024q^6}{\varepsilon}\cdot q^{4q^4/\varepsilon}\log n \le q^{42q^4/\varepsilon}\log n. ∣ S ∣ ≤ 2 q 2 C 1 8 q 2 / ε r 1 ≤ ε 1024 q 6 ⋅ q 4 q 4 / ε log n ≤ q 42 q 4 / ε log n . where we used
1024 q 6 ε ≤ q 14 / ε ≤ q 2 q 4 / ε . \frac{1024q^6}{\varepsilon}\le q^{14/\varepsilon}\le q^{2q^4/\varepsilon}. ε 1024 q 6 ≤ q 14/ ε ≤ q 2 q 4 / ε . Now, as t ≤ ∣ S ∣ t\le|S| t ≤ ∣ S ∣ d − ∣ A ∣ ≤ 2 ∣ S ∣ d-|A|\le2|S| d − ∣ A ∣ ≤ 2∣ S ∣ Corollary 5.1 that both
t + ∣ S ∣ + 1 ≤ C ∗ log n and C q ( d − ∣ A ∣ ) ≤ C ∗ log n , t+|S|+1\le C_*\log n \qquad\text{and}\qquad C_q(d-|A|)\le C_*\log n, t + ∣ S ∣ + 1 ≤ C ∗ log n and C q ( d − ∣ A ∣ ) ≤ C ∗ log n , due to C q ≤ q 5 q 2 C_q\le q^{5q^2} C q ≤ q 5 q 2 C ∗ C_* C ∗
C ∗ = q 48 q 4 / ε . C_* = q^{48q^4/\varepsilon}. C ∗ = q 48 q 4 / ε . By Lemma 5.1 , the set U 0 ⊆ U U_0\subseteq U U 0 ⊆ U
∣ U 0 ∣ ≥ ε n 16 ( C ∗ log n ) q 2 , |U_0|\ge\frac{\varepsilon n}{16(C_*\log n)^{q^2}}, ∣ U 0 ∣ ≥ 16 ( C ∗ log n ) q 2 ε n , so in Corollary 5.1 it suffices to ensure
n ≥ 16 ε ( C ∗ log n ) q 2 + 1 . n\ge\frac{16}{\varepsilon}(C_*\log n)^{q^2+1}. n ≥ ε 16 ( C ∗ log n ) q 2 + 1 . We will compute the logarithms of the righthand side expression, to estimate the value of n n n n 0 n_0 n 0
log n 0 ≥ log ( 16 / ε ) + ( q 2 + 1 ) log C ∗ + ( q 2 + 1 ) log log n 0 . \log n_0\ge\log(16/\varepsilon)+(q^2+1)\log C_*+(q^2+1)\log\log n_0. log n 0 ≥ log ( 16/ ε ) + ( q 2 + 1 ) log C ∗ + ( q 2 + 1 ) log log n 0 . Take
K : = q 6 log q ε , n 0 = 2 64 K . K:=\frac{q^6\log q}{\varepsilon},\qquad n_0=2^{64K}. K := ε q 6 log q , n 0 = 2 64 K . Since q ≥ 3 q\ge3 q ≥ 3 q 2 + 1 ≤ 5 4 q 2 q^2+1\le\frac{5}{4}q^2 q 2 + 1 ≤ 4 5 q 2
( q 2 + 1 ) log C ∗ ≤ 5 4 q 2 ⋅ 48 q 4 log q ε = 60 K . (q^2+1)\log C_*\le\frac{5}{4}q^2\cdot\frac{48q^4\log q}{\varepsilon}=60K. ( q 2 + 1 ) log C ∗ ≤ 4 5 q 2 ⋅ ε 48 q 4 log q = 60 K . In addition, log ( 16 / ε ) ≤ K \log(16/\varepsilon)\le K log ( 16/ ε ) ≤ K
log log n 0 = log ( 64 K ) = 6 + 6 log q + log log q + log ( 1 / ε ) ≤ 11 log q + log ( 1 / ε ) ≤ 12 log q ε , \log\log n_0=\log(64K)=6+6\log q+\log\log q+\log(1/\varepsilon)\le11\log q+\log(1/\varepsilon)\le\frac{12\log q}{\varepsilon}, log log n 0 = log ( 64 K ) = 6 + 6 log q + log log q + log ( 1/ ε ) ≤ 11 log q + log ( 1/ ε ) ≤ ε 12 log q , hence
( q 2 + 1 ) log log n 0 ≤ 15 q 2 log q ε ≤ K . (q^2+1)\log\log n_0\le\frac{15q^2\log q}{\varepsilon}\le K. ( q 2 + 1 ) log log n 0 ≤ ε 15 q 2 log q ≤ K . It follows that
log ( 16 ε ( C ∗ log n 0 ) q 2 + 1 ) ≤ K + 60 K + K ≤ 64 K = log n 0 . \log\left(\frac{16}{\varepsilon}(C_*\log n_0)^{q^2+1}\right)\le K+60K+K\le64K=\log n_0. log ( ε 16 ( C ∗ log n 0 ) q 2 + 1 ) ≤ K + 60 K + K ≤ 64 K = log n 0 . Clearly, n 0 ≥ max { q , 16 / ε } n_0\ge\max\{q,16/\varepsilon\} n 0 ≥ max { q , 16/ ε } n ≥ ⌈ n 0 ⌉ = ⌈ 2 64 K ⌉ n\ge\lceil n_0\rceil=\lceil2^{64K}\rceil n ≥ ⌈ n 0 ⌉ = ⌈ 2 64 K ⌉
