Introduction
Clustering with capacities models settings in which a center can handle only a bounded amount of work: a server hosts a limited number of jobs, a facility serves a limited number of customers, and an electoral district contains a bounded number of voters [31 , 36 ] . In the uniform capacitated ( k , z ) (k,z) ( k , z ) P P P n n n F F F k k k U U U k k k U U U z ≥ 1 z \ge1 z ≥ 1 z = 1 z = 1 z = 1 z = 2 z = 2 z = 2 k k k k k k
Without capacities, these problems are well understood in low-dimensional spaces: for every fixed z z z [14 , 71 , 35 , 57 , 32 ] . All these algorithms rely on locality. In an optimal solution, every client is served by a closest open center, so a region of the space interacts with the rest of the instance only through a few points near its boundary, and a local change of the solution affects only nearby clients. Capacities destroy this property. A client may have to travel to a far-away center because the nearby centers are full, and an algorithm must keep track of how many clients cross every boundary. As a result, much less is known. Arora, Raghavan and Rao [14 ] gave a quasi-polynomial-time approximation scheme for capacitated k k k 1 + ε 1 + \varepsilon 1 + ε [31 ] gave approximation schemes that respect uniform capacities, in quasi-polynomial time for k k k k k k k k k k k k
In general metrics, no constant-factor approximation is known even for capacitated k k k ( 2 − c ) k (2-c)k ( 2 − c ) k c > 0 c > 0 c > 0 [74 ] . A long line of work therefore violates the capacities by a factor of 1 + ε 1 + \varepsilon 1 + ε ( 1 + ε ) k (1 + \varepsilon)k ( 1 + ε ) k § 3 ). With ( 1 + ε ) k (1 + \varepsilon)k ( 1 + ε ) k k k k O ( ε − 2 log ( 1 / ε ) ) O(\varepsilon^{-2}\log(1/\varepsilon)) O ( ε − 2 log ( 1/ ε )) [74 , 75 ] . Lower bounds are known only for the uncapacitated problems [31 ] : k k k 1 + 2 / e 1 + 2/e 1 + 2/ e [68 ] , k k k [6 ] , under the unique games conjecture both are hard to approximate within some constant factor in high dimensions even with ( 1 + c ) k (1+c)k ( 1 + c ) k c > 0 c > 0 c > 0 [23 ] , and both are NP-hard in the plane [85 , 83 ] .
We study uniform capacitated ( k , z ) (k,z) ( k , z )
K : = k + ⌊ ε k ⌋ K := k + \lfloor\varepsilon k \rfloor K := k + ⌊ ε k ⌋ centers for a fixed ε ∈ ( 0 , 1 ] \varepsilon\in(0,1] ε ∈ ( 0 , 1 ] 2 k 2k 2 k U U U 2 k 2k 2 k § 2.1 ). The question is how fast the ratio grows as the number of extra centers decreases. In low dimensions, we give deterministic polynomial-time algorithms whose ratios stay bounded as ε → 0 \varepsilon\to0 ε → 0 k k k k k k [31 ] , these are the first polynomial-time algorithms with ( 1 + ε ) k (1 + \varepsilon)k ( 1 + ε ) k ε \varepsilon ε ε \varepsilon ε [16 , 65 ] or a constant factor larger capacities [44 BFRS15] . For soft capacities, our ratio is 3 z − ( 3 z − 1 ) ε 3^z - (3^z - 1)\varepsilon 3 z − ( 3 z − 1 ) ε 3 z 3^z 3 z 5 5 5 k k k ε \varepsilon ε ε = 1 \varepsilon= 1 ε = 1 § 2.3 ). For hard capacities, we convert these solutions. In general metrics, deleting centers greedily gives ratio 3 + 2 ln ( 1 / ε ) 3 + 2\ln(1/\varepsilon) 3 + 2 ln ( 1/ ε ) k k k ( 2 + o ( 1 ) ) z ln z ( 1 / ε ) (2 + o(1))^{z}\ln^{z}(1/\varepsilon) ( 2 + o ( 1 ) ) z ln z ( 1/ ε ) ε → 0 \varepsilon\to0 ε → 0 z z z ε \varepsilon ε
Our results
Our first result concerns Euclidean space, where centers may be placed anywhere. The dimension d d d
Theorem 1.1 (Euclidean space). For all fixed κ ≥ 1 \kappa\ge1 κ ≥ 1 z ≥ 1 z \ge1 z ≥ 1 ε ∈ ( 0 , 1 ] \varepsilon\in(0,1] ε ∈ ( 0 , 1 ] δ > 0 \delta> 0 δ > 0 n n n R d \mathbb{R}^{d} R d κ \kappa κ k k k U U U K K K R d \mathbb{R}^{d} R d O P T > 0 \mathrm{OPT} > 0 OPT > 0 U U U
( B z − ( B z − 1 ) ε + δ ) ⋅ O P T , where B z = min { u + 2 z v : ∣ 1 + t ∣ z ≤ u + g t + v ∣ t ∣ z for all t ∈ R } , \left(B_{z} - (B_{z} - 1)\varepsilon+ \delta\right)\cdot\mathrm{OPT}, \qquad\text{where } B_{z} = \min\{u + 2^{z}v : |1+t|^{z} \le u + gt + v|t|^{z}\text{ for all }t \in\mathbb{R}\}, ( B z − ( B z − 1 ) ε + δ ) ⋅ OPT , where B z = min { u + 2 z v : ∣1 + t ∣ z ≤ u + g t + v ∣ t ∣ z for all t ∈ R } , the minimum is over ( u , g , v ) ∈ R 3 (u,g,v) \in\mathbb{R}^{3} ( u , g , v ) ∈ R 3 O P T \mathrm{OPT} OPT k k k κ \kappa κ z z z ε \varepsilon ε δ \delta δ d d d
For k k k k k k B 1 = 3 B_{1} = 3 B 1 = 3 B 2 = 5 B_{2} = 5 B 2 = 5 3 − 2 ε + δ 3 - 2\varepsilon+ \delta 3 − 2 ε + δ 5 − 4 ε + δ 5 - 4\varepsilon+ \delta 5 − 4 ε + δ § 2.8 , we show that B z < 3 z B_{z} < 3^{z} B z < 3 z z > 1 z > 1 z > 1 Figure 1 (d)), that B z B_{z} B z
Figure 1. Our ratios without the additive δ \delta δ R d \mathbb{R}^d R d k k k R d \mathbb{R}^d R d F F F k k k Theorem 1.3 and the greedy bound of Theorem 1.4 with ρ = 1 \rho=1 ρ = 1 Theorem 1.4 ): 3 + 2 ln ( 1 / ε ) 3+2\ln(1/\varepsilon) 3 + 2 ln ( 1/ ε ) 7 + 2 ln ( 1 / ε ) 7+2\ln(1/\varepsilon) 7 + 2 ln ( 1/ ε ) k k k L 2 ( 9 , ε ) L_2(9,\varepsilon) L 2 ( 9 , ε ) L 2 ( 49 , ε ) L_2(49,\varepsilon) L 2 ( 49 , ε ) k k k ε → 0 \varepsilon\to0 ε → 0 R d \mathbb{R}^d R d 3 z 3^z 3 z B z B_z B z R d \mathbb{R}^d R d B z B_z B z B 1 = 3 B_1=3 B 1 = 3 B 2 = 5 B_2=5 B 2 = 5 B 3 ≈ 16.83 B_3\approx16.83 B 3 ≈ 16.83 B 4 ≈ 53.41 B_4\approx53.41 B 4 ≈ 53.41
Our other results concern a metric space in which the set F F F soft capacities, several centers may be opened at the same location; with hard capacities, every location hosts at most one center. In the theorems below, O P T \mathrm{OPT} OPT k k k
Theorem 1.2 (Soft capacities). For all fixed κ ≥ 1 \kappa\ge1 κ ≥ 1 z ≥ 1 z \ge1 z ≥ 1 ε ∈ ( 0 , 1 ] \varepsilon\in(0,1] ε ∈ ( 0 , 1 ] δ > 0 \delta> 0 δ > 0 P P P F F F κ \kappa κ K K K F F F U U U ( 3 z − ( 3 z − 1 ) ε + δ ) ⋅ O P T (3^{z} - (3^{z} - 1)\varepsilon+ \delta)\cdot\mathrm{OPT} ( 3 z − ( 3 z − 1 ) ε + δ ) ⋅ OPT
Theorem 1.3 (Hard capacities). For all fixed κ ≥ 1 \kappa\ge1 κ ≥ 1 z ≥ 1 z \ge1 z ≥ 1 ε ∈ ( 0 , 1 ] \varepsilon\in(0,1] ε ∈ ( 0 , 1 ] δ > 0 \delta> 0 δ > 0 Theorem 1.2 , outputs at most K K K F F F U U U
( ( 1 + 2 ( 3 z − ( 3 z − 1 ) ε ) 1 / z ) z + δ ) ⋅ O P T . \left((1 + 2(3^{z} - (3^{z} - 1)\varepsilon)^{1/z})^{z} + \delta\right)\cdot\mathrm{OPT}. ( ( 1 + 2 ( 3 z − ( 3 z − 1 ) ε ) 1/ z ) z + δ ) ⋅ OPT . For instances in R d \mathbb{R}^{d} R d κ \kappa κ F ⊂ R d F \subset\mathbb{R}^{d} F ⊂ R d 3 z 3^{z} 3 z B z B_{z} B z k k k ( 11 − 8 ε + 2 5 − 4 ε + δ ) ⋅ O P T (11 - 8\varepsilon+ 2\sqrt{5 - 4\varepsilon} + \delta)\cdot\mathrm{OPT} ( 11 − 8 ε + 2 5 − 4 ε + δ ) ⋅ OPT
For fixed z z z δ ≤ 1 / 2 \delta\le1/2 δ ≤ 1/2 P ∪ F P \cup F P ∪ F κ \kappa κ Theorems 1.2 and 1.3 run in time ( n + ∣ F ∣ ) κ / ( ε δ ) O ( κ / δ ) ( log Δ ) O ( 1 ) (n + |F|)^{\kappa/(\varepsilon\delta)^{O(\kappa/\delta)}}(\log\Delta)^{O(1)} ( n + ∣ F ∣ ) κ / ( ε δ ) O ( κ / δ ) ( log Δ ) O ( 1 ) Δ \Delta Δ P ∪ F P \cup F P ∪ F Theorem 1.1 , the exponent becomes ( 1 / ( ε δ ) ) 1 / δ O ( κ ) (1/(\varepsilon\delta))^{1/\delta^{O(\kappa)}} ( 1/ ( ε δ ) ) 1/ δ O ( κ ) § 8 ).
Our last result holds in every finite metric space and for every z ≥ 1 z \ge1 z ≥ 1 ρ ≥ 1 \rho\ge1 ρ ≥ 1 ε ∈ ( 0 , 1 ] \varepsilon\in(0,1] ε ∈ ( 0 , 1 ] L z ( ρ , ε ) ≥ ρ L_z(\rho,\varepsilon) \ge\rho L z ( ρ , ε ) ≥ ρ
∫ ρ L z ( ρ , ε ) d y ( y 1 / z + 2 ) z − y = ln 1 ε , \int_{\rho}^{L_z(\rho,\varepsilon)} \frac{dy}{(y^{1/z} + 2)^z - y} = \ln\frac{1}{\varepsilon}, ∫ ρ L z ( ρ , ε ) ( y 1/ z + 2 ) z − y d y = ln ε 1 , that is, L z ( ρ , ε ) L_z(\rho,\varepsilon) L z ( ρ , ε ) ln ( 1 / ε ) \ln(1/\varepsilon) ln ( 1/ ε ) y ′ = ( y 1 / z + 2 ) z − y y'=(y^{1/z}+2)^z-y y ′ = ( y 1/ z + 2 ) z − y y ( 0 ) = ρ y(0)=\rho y ( 0 ) = ρ L 1 ( ρ , ε ) = ρ + 2 ln ( 1 / ε ) L_1(\rho,\varepsilon)=\rho+2\ln(1/\varepsilon) L 1 ( ρ , ε ) = ρ + 2 ln ( 1/ ε ) s = L 2 ( ρ , ε ) 1 / 2 s=L_2(\rho,\varepsilon)^{1/2} s = L 2 ( ρ , ε ) 1/2 s − ln ( 1 + s ) = ρ − ln ( 1 + ρ ) + 2 ln ( 1 / ε ) s-\ln(1+s)=\sqrt{\rho}-\ln(1+\sqrt{\rho})+2\ln(1/\varepsilon) s − ln ( 1 + s ) = ρ − ln ( 1 + ρ ) + 2 ln ( 1/ ε ) z z z ρ \rho ρ L z ( ρ , ε ) = ( 2 + o ( 1 ) ) z ln z ( 1 / ε ) L_z(\rho,\varepsilon)=(2+o(1))^z\ln^z(1/\varepsilon) L z ( ρ , ε ) = ( 2 + o ( 1 ) ) z ln z ( 1/ ε ) ε → 0 \varepsilon\to0 ε → 0 § 11.2 ).
Theorem 1.4 (General metrics). Let z ≥ 1 z\ge1 z ≥ 1 ρ ≥ 1 \rho\ge1 ρ ≥ 1 ε ∈ ( 0 , 1 ] \varepsilon\in(0,1] ε ∈ ( 0 , 1 ] δ > 0 \delta>0 δ > 0 k k k F F F ρ \rho ρ K K K F F F U U U ( L z ( ρ , ε ) + δ ) ⋅ O P T (L_z(\rho,\varepsilon)+\delta)\cdot\mathrm{OPT} ( L z ( ρ , ε ) + δ ) ⋅ OPT ( L z ( ( 1 + 2 ρ 1 / z ) z , ε ) + δ ) ⋅ O P T (L_z((1+2\rho^{1/z})^z,\varepsilon)+\delta)\cdot\mathrm{OPT} ( L z (( 1 + 2 ρ 1/ z ) z , ε ) + δ ) ⋅ OPT
In every finite metric space, local search gives ρ = 3 + δ \rho=3+\delta ρ = 3 + δ k k k [9 ] , so Theorem 1.4 gives the ratios 3 + 2 ln ( 1 / ε ) + δ 3+2\ln(1/\varepsilon)+\delta 3 + 2 ln ( 1/ ε ) + δ 7 + 2 ln ( 1 / ε ) + δ 7+2\ln(1/\varepsilon)+\delta 7 + 2 ln ( 1/ ε ) + δ [34 ] gives 2 + 2 ln ( 1 / ε ) + δ 2+2\ln(1/\varepsilon)+\delta 2 + 2 ln ( 1/ ε ) + δ 5 + 2 ln ( 1 / ε ) + δ 5+2\ln(1/\varepsilon)+\delta 5 + 2 ln ( 1/ ε ) + δ k k k ρ = 9 + δ \rho=9+\delta ρ = 9 + δ [13 ] or ρ = 4.9 + δ \rho=4.9+\delta ρ = 4.9 + δ z z z ρ \rho ρ [62 ] . In the settings of Theorems 1.2 and 1.3 , we can take ρ = 1 + δ \rho=1+\delta ρ = 1 + δ § 11.2 ). For hard capacities, this gives the ratio L z ( 3 z , ε ) + δ L_z(3^z,\varepsilon)+\delta L z ( 3 z , ε ) + δ Theorem 1.3 for 0.21 < ε < 1 0.21<\varepsilon<1 0.21 < ε < 1 k k k 0.27 < ε < 1 0.27<\varepsilon<1 0.27 < ε < 1 k k k
Table 1 compares our ratios with previous results. Theorem 1.4 improves the ratio O ( ε − 2 log ( 1 / ε ) ) O(\varepsilon^{-2}\log(1/\varepsilon)) O ( ε − 2 log ( 1/ ε )) [74 , 75 ] for k k k ( 1 + ε ) k (1+\varepsilon)k ( 1 + ε ) k 3 + 2 ln ( 1 / ε ) 3+2\ln(1/\varepsilon) 3 + 2 ln ( 1/ ε ) 7 + 2 ln ( 1 / ε ) 7+2\ln(1/\varepsilon) 7 + 2 ln ( 1/ ε ) k k k Figure 1 shows how our ratios depend on ε \varepsilon ε z z z
Table 1. Approximation ratios of our algorithms (Theorems 1.1 to 1.4 ), with K = k + ⌊ ε k ⌋ K=k+\lfloor\varepsilon k\rfloor K = k + ⌊ ε k ⌋ ε \varepsilon ε δ > 0 \delta>0 δ > 0 n O ( 1 ) n^{O(1)} n O ( 1 ) n polylog ( n ) n^{\operatorname{polylog}(n)} n polylog ( n ) f ( k , δ ) ⋅ n O ( 1 ) f(k,\delta)\cdot n^{O(1)} f ( k , δ ) ⋅ n O ( 1 ) R d \mathbb{R}^{d} R d F F F Theorem 1.4 with ρ = 1 \rho=1 ρ = 1 ρ z \rho_{z} ρ z ( k , z ) (k,z) ( k , z ) ρ 1 = 3 \rho_{1}=3 ρ 1 = 3 [9 ] and ρ 2 = 9 \rho_{2}=9 ρ 2 = 9 [13 ] ; the randomized algorithm of [34 ] gives ρ 1 = 2 \rho_{1}=2 ρ 1 = 2 ρ 2 = 4.9 \rho_{2}=4.9 ρ 2 = 4.9 ε = 1 / 2 \varepsilon=1/2 ε = 1/2 L 2 ( 9 , ε ) ≈ 22.6 L_{2}(9,\varepsilon)\approx22.6 L 2 ( 9 , ε ) ≈ 22.6 L 2 ( 49 , ε ) ≈ 73.4 L_{2}(49,\varepsilon)\approx73.4 L 2 ( 49 , ε ) ≈ 73.4
Our techniques
We split the clusters of a near-optimal uncapacitated solution into at most 2 k − 1 2k-1 2 k − 1 U U U ( u , v ) (u,v) ( u , v ) movable if its clients can be assigned to any destinations of its mass at cost at most u u u v v v β = u + 2 z v \beta=u+2^{z}v β = u + 2 z v K K K rescue blocks: a rescued block gets a private center and does not move, which saves its share of the cost above the internal cost of all blocks. Rescuing blocks one at a time saves only an ε / 2 \varepsilon/2 ε /2 2 k 2k 2 k excess , the number of its blocks minus the number of its centers. As there are at most 2 k − 1 2k-1 2 k − 1 k k k rescue lemma (Theorem 2.5) shows that ⌊ ε k ⌋ \lfloor\varepsilon k\rfloor ⌊ ε k ⌋ ε \varepsilon ε β − ( β − 1 ) ε \beta-(\beta-1)\varepsilon β − ( β − 1 ) ε
The extra centers also make polynomial time possible. The algorithm does not know the optimal solution, so it optimizes over all ways of routing the blocks, by a dynamic program over a network built from a hierarchical decomposition. With a constant number of portals per cell, the network is accurate except for pairs of points that the decomposition cuts badly, and a reserved part of the extra centers rescues the few blocks on such pairs, so that the dynamic program runs in polynomial time. In Euclidean space, a random matching between the clients of a block and the destinations of its mass cancels a first-order term, so that movability reduces to the one-dimensional inequality in Theorem 1.1 . Theorems 1.1 and 1.2 are instances of one framework theorem (Theorem 2.6 ) for block partitions in which every block is ( u , v ) (u,v) ( u , v ) β = 3 z \beta= 3^{z} β = 3 z β = B z \beta= B_{z} β = B z Theorem 1.3 adds a conversion from soft to hard capacities due to Li [74 ] , which we extend to all z z z projection lemma ).
In general metrics, there is no decomposition to optimize over, and Theorem 1.4 is not an instance of the framework theorem. It starts from one center per block, converted to hard capacities by the same lemma if needed, and deletes centers greedily, as the reverse greedy algorithm [46 , 8 ] does starting from all locations. The cost is supermodular in the set of centers, and the projection lemma, applied to an optimal solution, shows that for k k k k + r k+r k + r 2 O P T / r 2\mathrm{OPT}/r 2 OPT / r 2 k 2k 2 k K K K 2 ln ( 1 / ε ) O P T 2\ln(1/\varepsilon)\mathrm{OPT} 2 ln ( 1/ ε ) OPT log n \log n log n z > 1 z>1 z > 1 L z L_z L z
Organization. § 2 explains our approach: blocks and the rescue lemma (§ 2.1 to § 2.3 ), the network and the framework theorem (§ 2.4 to § 2.6 ), and the extensions (§ 2.7 to § 2.9 ). After related work (§ 3 ) and preliminaries (§ 4 ), § 5 to § 8 prove the framework theorem, § 9 to § 11 prove our results, and § 12 lists open questions; the appendices characterize B z B_z B z
Our Approach
Setting. An instance consists of a metric space ( V , dist ) (V,\operatorname{dist}) ( V , dist ) F ⊆ V F \subseteq V F ⊆ V locations , which is finite except in the Euclidean setting below, a multiset P P P n n n clients at points of V V V k k k U U U z ≥ 1 z \ge1 z ≥ 1 solution consists of centers, each placed at a location in F F F σ \sigma σ U U U cost ( σ ) = ∑ p ∈ P dist ( p , σ ( p ) ) z \operatorname{cost}(\sigma)=\sum_{p\in P}\operatorname{dist}(p,\sigma(p))^{z} cost ( σ ) = ∑ p ∈ P dist ( p , σ ( p ) ) z dist ( p , σ ( p ) ) \operatorname{dist}(p,\sigma(p)) dist ( p , σ ( p )) p p p σ ( p ) \sigma(p) σ ( p ) soft capacities, several centers may be placed at the same location; with hard capacities, the centers are at distinct locations. In the Euclidean setting , V = F = R d V=F=\mathbb{R}^{d} V = F = R d ∥ ⋅ ∥ \lVert\cdot\rVert ∥ ⋅ ∥ O P T \mathrm{OPT} OPT k k k O P T \mathrm{OPT} OPT O P T > 0 \mathrm{OPT}>0 OPT > 0 k < n k<n k < n k k k n ≤ k U n\leq kU n ≤ k U κ \kappa κ 2 κ 2^{\kappa} 2 κ § 4 collects this and other standard facts.
This section explains the algorithms and their analysis. We first consider soft capacities in a metric space with a finite set F F F P ∪ F P\cup F P ∪ F κ \kappa κ § 2.7 and § 2.8 describe the changes for one-sided low dimension and for Euclidean space, and § 2.9 treats hard capacities and general metrics. The definitions, lemmas and theorems stated in this section are used in the rest of the paper, while the network, the decompositions and the tools for the extensions are described informally here and defined in § 6 , § 7 and § 9 to § 11 . The accuracy parameter η ∈ ( 0 , 1 / 10 ] \eta\in(0,1/10] η ∈ ( 0 , 1/10 ] δ \delta δ δ / 3 \delta/3 δ /3 (9) in § 8 , depend on κ \kappa κ z z z ε \varepsilon ε η \eta η
Algorithm 1 Soft capacities, P ∪ F P \cup F P ∪ F P P P F F F k k k U U U κ \kappa κ z z z ε \varepsilon ε η \eta η ( u , v ) = ( 3 z − 1 , ( 3 / 2 ) z − 1 ) (u,v) = (3^{z-1},(3/2)^{z-1}) ( u , v ) = ( 3 z − 1 , ( 3/2 ) z − 1 )
If k < k 0 k < k_0 k < k 0 k 0 = O ( 1 / η 2 ) k_0 = O(1/\eta^2) k 0 = O ( 1/ η 2 ) (9) , try every multiset of at most k k k
Compute an uncapacitated solution with at most k k k F F F ( 1 + η ) OPT (1+\eta)\operatorname{OPT} ( 1 + η ) OPT Theorem 4.3 ), and split its clusters into blocks of at most U U U § 2.1 ).
For every hierarchical decomposition D \mathcal{D} D H \mathcal{H} H N D \mathcal{N}_{\mathcal{D}} N D § 2.4 and § 2.5 ).
Turn the cheapest network solution found into a clustering: match the clients of every normal block to the end points of its paths, serve every rescued block by a center at its representative (Theorem 7.5), and return the result.
Blocks and movable blocks
Reassigning every client to a closest center turns a solution with at most k k k Theorem 4.3 gives an uncapacitated solution with at most k k k F F F ( 1 + η ) OPT (1+\eta)\operatorname{OPT} ( 1 + η ) OPT Q Q Q ⌈ ∣ Q ∣ / U ⌉ \lceil|Q|/U\rceil ⌈ ∣ Q ∣/ U ⌉ U U U
Definition 2.1 (Block partition). A block partition of P P P P P P P 1 , … , P m P_1,\ldots,P_m P 1 , … , P m m ≤ 2 k − 1 m \le2k-1 m ≤ 2 k − 1 w i : = ∣ P i ∣ ≤ U w_i := |P_i| \le U w i := ∣ P i ∣ ≤ U i i i representative a i ∈ F a_i \in F a i ∈ F internal cost of block i i i A i : = ∑ p ∈ P i dist ( p , a i ) z A_i := \sum_{p\in P_i}\operatorname{dist}(p,a_i)^z A i := ∑ p ∈ P i dist ( p , a i ) z A : = ∑ i A i A := \sum_i A_i A := ∑ i A i
If the representative of a block is the center of its cluster, then A A A A ≤ ( 1 + η ) OPT A \le(1+\eta)\operatorname{OPT} A ≤ ( 1 + η ) OPT n ≤ k U n \le kU n ≤ k U k k k
m = ∑ Q ⌈ ∣ Q ∣ U ⌉ < ∑ Q ( ∣ Q ∣ U + 1 ) ≤ 2 k . (1) m = \sum_Q \left\lceil\frac{|Q|}{U}\right\rceil< \sum_Q \left(\frac{|Q|}{U}+1\right) \le2k. \tag*{(1)} m = Q ∑ ⌈ U ∣ Q ∣ ⌉ < Q ∑ ( U ∣ Q ∣ + 1 ) ≤ 2 k . (1) So at most 2 k − 1 2k-1 2 k − 1 A ≤ ( 1 + η ) OPT A \le(1+\eta)\operatorname{OPT} A ≤ ( 1 + η ) OPT § 2.8 ). With K K K m − K m-K m − K w i w_i w i a i a_i a i
Definition 2.2 (Movable block). Let u ≥ 1 u \ge1 u ≥ 1 v ≥ 0 v \ge0 v ≥ 0 i i i ( u , v ) (u,v) ( u , v ) y 1 , … , y w i ∈ F y_1,\ldots,y_{w_i} \in F y 1 , … , y w i ∈ F ϕ : P i → { 1 , … , w i } \phi:P_i \to\{1,\ldots,w_i\} ϕ : P i → { 1 , … , w i }
∑ p ∈ P i dist ( p , y ϕ ( p ) ) z ≤ u ⋅ A i + v ⋅ ∑ r = 1 w i dist ( a i , y r ) z . \sum_{p\in P_i}\operatorname{dist}(p,y_{\phi(p)})^z \le u \cdot A_i + v \cdot\sum_{r=1}^{w_i}\operatorname{dist}(a_i,y_r)^z. p ∈ P i ∑ dist ( p , y ϕ ( p ) ) z ≤ u ⋅ A i + v ⋅ r = 1 ∑ w i dist ( a i , y r ) z . We think of y 1 , … , y w i y_1,\ldots,y_{w_i} y 1 , … , y w i
Lemma 2.3 (Triangle bound). Every block of every block partition is ( 3 z − 1 , ( 3 / 2 ) z − 1 ) (3^{z-1},(3/2)^{z-1}) ( 3 z − 1 , ( 3/2 ) z − 1 )
Proof. For r , s ≥ 0 r,s\ge0 r , s ≥ 0
( r + s ) z = ( 1 3 ⋅ 3 r + 2 3 ⋅ 3 2 s ) z ≤ 3 z − 1 r z + ( 3 2 ) z − 1 s z . (r+s)^z=\left(\frac{1}{3}\cdot3r+\frac{2}{3}\cdot\frac{3}{2}s\right)^z\le3^{z-1}r^z+\left(\frac{3}{2}\right)^{z-1}s^z. ( r + s ) z = ( 3 1 ⋅ 3 r + 3 2 ⋅ 2 3 s ) z ≤ 3 z − 1 r z + ( 2 3 ) z − 1 s z . Apply this to r = dist ( p , a i ) r=\operatorname{dist}(p,a_i) r = dist ( p , a i ) s = dist ( a i , y ϕ ( p ) ) s=\operatorname{dist}(a_i,y_{\phi(p)}) s = dist ( a i , y ϕ ( p ) ) ϕ \phi ϕ p ∈ P i p\in P_i p ∈ P i
Following a reference solution
The analysis compares the blocks with a reference solution S ∗ S^* S ∗ k k k F F F J J J S ∗ S^* S ∗ ∣ J ∣ ≤ k |J|\le k ∣ J ∣ ≤ k j ∈ J j\in J j ∈ J c j c_j c j ℓ j ≤ U \ell_j\le U ℓ j ≤ U I = { 1 , … , m } I=\{1,\ldots,m\} I = { 1 , … , m }
Lemma 2.4 (Transport forest). There is an integral transport plan π ∈ Z ≥ 0 I × J \pi\in\mathbb{Z}_{\ge0}^{I\times J} π ∈ Z ≥ 0 I × J w i w_i w i i ∈ I i\in I i ∈ I ℓ j \ell_j ℓ j j ∈ J j\in J j ∈ J T : = ∑ i , j π i j dist ( a i , c j ) z T:=\sum_{i,j}\pi_{ij}\operatorname{dist}(a_i,c_j)^z T := ∑ i , j π ij dist ( a i , c j ) z
T ≤ ( A 1 / z + cost ( S ∗ ) 1 / z ) z ≤ 2 z − 1 ( A + cost ( S ∗ ) ) . T\le\left(A^{1/z}+\operatorname{cost}(S^*)^{1/z}\right)^z\le2^{z-1}\left(A+\operatorname{cost}(S^*)\right). T ≤ ( A 1/ z + cost ( S ∗ ) 1/ z ) z ≤ 2 z − 1 ( A + cost ( S ∗ ) ) . Proof. Let π i j 0 \pi^0_{ij} π ij 0 P i P_i P i S ∗ S^* S ∗ j j j (6) to the triples ( a i , c j , p ) (a_i,c_j,p) ( a i , c j , p ) p p p p ∈ P i p\in P_i p ∈ P i j j j π 0 \pi^0 π 0 ( A 1 / z + cost ( S ∗ ) 1 / z ) z \left(A^{1/z}+\operatorname{cost}(S^*)^{1/z}\right)^z ( A 1/ z + cost ( S ∗ ) 1/ z ) z π 0 \pi^0 π 0
We fix such a plan π \pi π π i j \pi_{ij} π ij i i i a i a_i a i c j c_j c j T i : = ∑ j π i j dist ( a i , c j ) z T_i:=\sum_j\pi_{ij}\operatorname{dist}(a_i,c_j)^z T i := ∑ j π ij dist ( a i , c j ) z moving cost of block i i i π \pi π i i i π i j \pi_{ij} π ij j j j j j j S ∗ S^* S ∗ u A + v T uA+vT u A + v T
β : = u + 2 z v , \beta:=u+2^zv, β := u + 2 z v , Theorem 2.4 gives, if A ≤ ( 1 + η ) cost ( S ∗ ) A\le(1+\eta)\operatorname{cost}(S^*) A ≤ ( 1 + η ) cost ( S ∗ )
u A + v T ≤ ( u + 2 z − 1 v ) A + 2 z − 1 v cost ( S ∗ ) ≤ β ( 1 + η ) cost ( S ∗ ) . (2) uA+vT\le(u+2^{z-1}v)A+2^{z-1}v\operatorname{cost}(S^*)\le\beta(1+\eta)\operatorname{cost}(S^*). \tag*{(2)} u A + v T ≤ ( u + 2 z − 1 v ) A + 2 z − 1 v cost ( S ∗ ) ≤ β ( 1 + η ) cost ( S ∗ ) . (2) With an optimal S ∗ S^* S ∗ ε → 0 \varepsilon\to0 ε → 0 β = 3 z − 1 + 2 z ( 3 / 2 ) z − 1 = 3 z \beta=3^{z-1}+2^z(3/2)^{z-1}=3^z β = 3 z − 1 + 2 z ( 3/2 ) z − 1 = 3 z
Rescuing blocks
An extra center can rescue a block: a new center at a i a_i a i P i P_i P i w i ≤ U w_i\le U w i ≤ U A i A_i A i u A i + v T i uA_i+vT_i u A i + v T i i i i (2) . So rescuing block i i i (2) by its saving
W i : = ( u − 1 ) A i + v T i ≥ 0 , W_i:=(u-1)A_i+vT_i\ge0, W i := ( u − 1 ) A i + v T i ≥ 0 , and by Theorem 2.4, if A ≤ ( 1 + η ) cost ( S ∗ ) A\le(1+\eta)\operatorname{cost}(S^*) A ≤ ( 1 + η ) cost ( S ∗ )
∑ i W i = ( u − 1 ) A + v T ≤ ( u − 1 + 2 z − 1 v ) A + 2 z − 1 v cost ( S ∗ ) ≤ ( β − 1 ) ( 1 + η ) cost ( S ∗ ) . (3) \sum_i W_i=(u-1)A+vT\le(u-1+2^{z-1}v)A+2^{z-1}v\operatorname{cost}(S^*)\le(\beta-1)(1+\eta)\operatorname{cost}(S^*). \tag*{(3)} i ∑ W i = ( u − 1 ) A + v T ≤ ( u − 1 + 2 z − 1 v ) A + 2 z − 1 v cost ( S ∗ ) ≤ ( β − 1 ) ( 1 + η ) cost ( S ∗ ) . (3) Suppose that ⌊ ε k ⌋ \lfloor\varepsilon k\rfloor ⌊ ε k ⌋ ( ε − η ) ∑ i W i (\varepsilon-\eta)\sum_i W_i ( ε − η ) ∑ i W i
A + ( 1 − ε + η ) ∑ i W i ≤ ( β − ( β − 1 ) ε + O ( β η ) ) ⋅ cost ( S ∗ ) . (4) A+(1-\varepsilon+\eta)\sum_i W_i\le\left(\beta-(\beta-1)\varepsilon+O(\beta\eta)\right)\cdot\operatorname{cost}(S^*). \tag*{(4)} A + ( 1 − ε + η ) i ∑ W i ≤ ( β − ( β − 1 ) ε + O ( β η ) ) ⋅ cost ( S ∗ ) . (4) which is the ratio of our theorems.
The obstacle. The natural strategy rescues the ⌊ ε k ⌋ \lfloor\varepsilon k\rfloor ⌊ ε k ⌋ m m m 2 k 2k 2 k ε / 2 \varepsilon/2 ε /2 ∑ i W i \sum_i W_i ∑ i W i β − ( β − 1 ) ε / 2 \beta-(\beta-1)\varepsilon/2 β − ( β − 1 ) ε /2
Closing centers. Rescuing blocks also removes clients from the centers of S ∗ S^* S ∗ j j j j j j Figure 2 ). The support of π \pi π m m m ∣ J ∣ |J| ∣ J ∣ excess , the number of its blocks minus the number of its centers. Consider a budget of b ≤ k b\le k b ≤ k b b b ε k \varepsilon k ε k m − ∣ J ∣ ≤ 2 k − 1 − ∣ J ∣ m-|J|\le2k-1-|J| m − ∣ J ∣ ≤ 2 k − 1 − ∣ J ∣ ∣ J ∣ |J| ∣ J ∣ S ∗ S^* S ∗ k − ∣ J ∣ k-|J| k − ∣ J ∣ S ∗ S^* S ∗ b + k − ∣ J ∣ b+k-|J| b + k − ∣ J ∣
Figure 2. The support of the transport plan π \pi π S ∗ S^* S ∗
b + k − ∣ J ∣ 2 k − ∣ J ∣ ≥ b k (which is equivalent to ( k − ∣ J ∣ ) ( k − b ) ≥ 0 ), \frac{b+k-|J|}{2k-|J|}\ge\frac{b}{k}\qquad\text{(which is equivalent to }(k-|J|)(k-b)\ge0\text{),} 2 k − ∣ J ∣ b + k − ∣ J ∣ ≥ k b (which is equivalent to ( k − ∣ J ∣ ) ( k − b ) ≥ 0 ), rescuing this fraction of every component, if that were possible, would gain a b / k b/k b / k 3 k / ( L + 1 ) 3k/(L+1) 3 k / ( L + 1 ) 3 k 3k 3 k L L L Figure 2 (b)). Then we rescue all components of nonpositive excess, which cost nothing or even free centers, and add the other components greedily in order of saving per unit of excess; the greedy choice wastes at most one component, of excess at most L L L rescue lemma shows that this loses only O ( L / k + 1 / L ) O(L/k+1/L) O ( L / k + 1/ L ) § 5 .
Lemma 2.5 (Rescue lemma). Let π \pi π integral transport plan between disjoint index sets I I I J J J ∣ I ∣ ≤ 2 k − 1 |I|\le2k-1 ∣ I ∣ ≤ 2 k − 1 ∣ J ∣ ≤ k |J|\le k ∣ J ∣ ≤ k W i ≥ 0 W_i\ge0 W i ≥ 0 i ∈ I i\in I i ∈ I L ≥ 1 L\ge1 L ≥ 1 0 ≤ b ≤ k 0\le b\le k 0 ≤ b ≤ k R ⊆ I R\subseteq I R ⊆ I J 0 ⊆ J J_0\subseteq J J 0 ⊆ J
(i) π i j = 0 \pi_{ij}=0 π ij = 0 i ∈ I ∖ R i\in I\setminus R i ∈ I ∖ R j ∈ J 0 j\in J_0 j ∈ J 0
(ii) ∣ R ∣ + ∣ J ∣ − ∣ J 0 ∣ ≤ k + b |R|+|J|-|J_0|\le k+b ∣ R ∣ + ∣ J ∣ − ∣ J 0 ∣ ≤ k + b
(iii) ∑ i ∈ R W i ≥ ( b − L k − 3 L + 1 ) ∑ i ∈ I W i \sum_{i\in R}W_i\ge\left(\frac{b-L}{k}-\frac{3}{L+1}\right)\sum_{i\in I}W_i ∑ i ∈ R W i ≥ ( k b − L − L + 1 3 ) ∑ i ∈ I W i
By (i), rescuing the blocks in R R R J 0 J_0 J 0 k + b k+b k + b L ≈ 12 / η L\approx12/\eta L ≈ 12/ η b ≈ ε k b\approx\varepsilon k b ≈ ε k k k k L L L ( ε − η ) ∑ i W i (\varepsilon-\eta)\sum_iW_i ( ε − η ) ∑ i W i (4) . Small values of k k k
Finding a solution: the network and the dynamic program
The rescued solution is defined in terms of the unknown solution S ∗ S^* S ∗ hierarchical decomposition D \mathcal{D} D X = P ∪ F X = P \cup F X = P ∪ F X X X cells at levels 0 , … , H 0,\ldots,H 0 , … , H h h h 2 h 2^h 2 h h + 1 h+1 h + 1 0 0 0 X X X H H H H = O ( log Δ ( X ) ) H = O(\log\Delta(X)) H = O ( log Δ ( X )) Δ ( X ) \Delta(X) Δ ( X ) X X X
The network. Every cell C C C h h h N C N_C N C portals , a maximal subset of C C C λ 2 h \lambda2^h λ 2 h ∣ N C ∣ ≤ M : = ⌈ ( 4 / λ ) κ ⌉ \lvert N_C\rvert\le M := \lceil(4/\lambda)^\kappa\rceil ∣ N C ∣ ≤ M := ⌈( 4/ λ ) κ ⌉ ascending and a descending copy (Figure 3 and Theorem 7.1). A unit of mass travels from a point up the hierarchy along ascending copies, each joined by an arc of cost 0 0 0 λ 2 h + 1 \lambda2^{h+1} λ 2 h + 1 z z z crossing arc from the ascending copy of a portal x ′ x' x ′ y ′ y' y ′ ( dist ( x ′ , y ′ ) + 4 λ 2 h ) z (\operatorname{dist}(x',y') + 4\lambda2^h)^z ( dist ( x ′ , y ′ ) + 4 λ 2 h ) z 2 λ 2 h 2\lambda2^h 2 λ 2 h x x x y y y dist ( x , y ) z \operatorname{dist}(x,y)^z dist ( x , y ) z the network never underestimates distances . We call distinct points x , y x,y x , y Λ \Lambda Λ badly cut , if the lowest cell that contains both has a level h h h 2 h > Λ ⋅ dist ( x , y ) 2^h > \Lambda\cdot\operatorname{dist}(x,y) 2 h > Λ ⋅ dist ( x , y ) ( x , y ) (x,y) ( x , y ) Λ \Lambda Λ x x x y y y ( ( 1 + 8 λ Λ ) dist ( x , y ) ) z ((1+8\lambda\Lambda)\operatorname{dist}(x,y))^z (( 1 + 8 λ Λ ) dist ( x , y ) ) z
Figure 3. Part of the network: a cell C C C h h h x x x x ′ x' x ′ x x x y y y y ′ y' y ′ y y y ( dist ( x ′ , y ′ ) + 4 λ 2 h ) z (\operatorname{dist}(x',y')+4\lambda2^h)^z ( dist ( x ′ , y ′ ) + 4 λ 2 h ) z dist ( x , y ) z \operatorname{dist}(x,y)^z dist ( x , y ) z
Network solutions and the dynamic program. A network solution declares every block either rescued or normal , routes the mass of the normal blocks from their representatives to locations, and opens ⌈ t / U ⌉ \lceil t/U\rceil ⌈ t / U ⌉ t t t K K K
Φ = ∑ i n o r m a l ( u − 1 ) A i + v ∑ a r c s e cost ( e ) f ( e ) , \Phi= \sum_{i\ \mathrm{normal}} (u-1)A_i + v \sum_{\mathrm{arcs}\ e} \operatorname{cost}(e)f(e), Φ = i normal ∑ ( u − 1 ) A i + v arcs e ∑ cost ( e ) f ( e ) , where f f f K K K A + Φ A+\Phi A + Φ O ( M ) O(M) O ( M ) n O ( M ) n^{O(M)} n O ( M ) λ \lambda λ
Badly cut pairs and the extra centers
The network is accurate only for pairs that are not Λ \Lambda Λ λ \lambda λ M M M [14 , 90 ] , every pair pays for the error of the level at which it is cut, and keeping the total error small makes the number of portals grow with the number of levels. We use extra centers instead, as suggested by Cohen-Addad [31 ] . Call an edge i j ij ij supp ( π ) \operatorname{supp}(\pi) supp ( π ) ( a i , c j ) (a_i,c_j) ( a i , c j ) Λ \Lambda Λ Λ \Lambda Λ O ( κ / Λ ) O(\kappa/\Lambda) O ( κ /Λ ) [61 , 90 ] . We derandomize this (§ 6 ): at every level h h h t 2 h t2^h t 2 h net , a maximal set of points that are pairwise more than 2 h − 3 2^{h-3} 2 h − 3 t ∈ [ 1 / 4 , 1 / 2 ] t \in[1/4,1/2] t ∈ [ 1/4 , 1/2 ] t t t O ( ∣ X ∣ 2 ) O(|X|^2) O ( ∣ X ∣ 2 ) H \mathcal{H} H Λ \Lambda Λ 2 11 κ / Λ 2^{11}\kappa/\Lambda 2 11 κ /Λ H \mathcal{H} H 3 k 3k 3 k H \mathcal{H} H 3 ⋅ 2 11 κ k / Λ 3 \cdot2^{11}\kappa k/\Lambda 3 ⋅ 2 11 κk /Λ ⌊ γ k ⌋ \lfloor\gamma k \rfloor ⌊ γ k ⌋ Λ = 2 13 κ / γ \Lambda= 2^{13}\kappa/\gamma Λ = 2 13 κ / γ γ = min { ϵ , η } / 4 \gamma= \min\{\epsilon,\eta\}/4 γ = min { ϵ , η } /4 ⌊ γ k ⌋ \lfloor\gamma k \rfloor ⌊ γ k ⌋ 1 + 8 λ Λ 1 + 8\lambda\Lambda 1 + 8 λ Λ λ = η / ( 8 z Λ ) \lambda= \eta/(8z\Lambda) λ = η / ( 8 z Λ ) 1 + η / z 1 + \eta/z 1 + η / z ( 1 + η / z ) z ≤ e η (1+\eta/z)^z \le e^\eta ( 1 + η / z ) z ≤ e η λ \lambda λ M M M
The framework theorem
To summarize, the algorithm reserves ⌊ γ k ⌋ \lfloor\gamma k \rfloor ⌊ γ k ⌋ b = ⌊ ϵ k ⌋ − ⌊ γ k ⌋ b = \lfloor\epsilon k \rfloor- \lfloor\gamma k \rfloor b = ⌊ ϵ k ⌋ − ⌊ γ k ⌋ L = Θ ( 1 / η ) L = \Theta(1/\eta) L = Θ ( 1/ η ) k < k 0 = Θ ( 1 / η 2 ) k < k_0 = \Theta(1/\eta^2) k < k 0 = Θ ( 1/ η 2 ) 1 + η / z 1+\eta/z 1 + η / z § 8 collects the parameters in (9) and proves the following theorem.
Theorem 2.6 (Framework theorem). Fix κ ≥ 1 \kappa\ge1 κ ≥ 1 z ≥ 1 z \ge1 z ≥ 1 ϵ ∈ ( 0 , 1 ] \epsilon\in(0,1] ϵ ∈ ( 0 , 1 ] η ∈ ( 0 , 1 / 10 ] \eta\in(0,1/10] η ∈ ( 0 , 1/10 ] u ≥ 1 u \ge1 u ≥ 1 v ≥ 0 v \ge0 v ≥ 0 β = u + 2 z v \beta= u + 2^zv β = u + 2 z v P ∪ F P \cup F P ∪ F κ \kappa κ ( u , v ) (u,v) ( u , v ) K K K
( β − ( β − 1 ) ϵ + 4 β η ) ⋅ cost ( S ∗ ) \bigl(\beta-(\beta-1)\epsilon+4\beta\eta\bigr)\cdot\operatorname{cost}(S^*) ( β − ( β − 1 ) ϵ + 4 β η ) ⋅ cost ( S ∗ ) for every soft solution S ∗ S^* S ∗ k k k A ≤ ( 1 + η ) cost ( S ∗ ) A \le(1+\eta)\operatorname{cost}(S^*) A ≤ ( 1 + η ) cost ( S ∗ ) n n n ∣ F ∣ |F| ∣ F ∣ log Δ ( P ∪ F ) \log\Delta(P \cup F) log Δ ( P ∪ F ) κ \kappa κ z z z ϵ \epsilon ϵ η \eta η
The guarantee holds for all such S ∗ S^* S ∗ S ∗ S^* S ∗ § 10 uses a reference solution that is not optimal. With the block partition of § 2.1 , Theorem 2.3 and an optimal soft solution as S ∗ S^* S ∗ Theorem 2.6 gives Theorem 1.2 when P ∪ F P \cup F P ∪ F κ \kappa κ
One-sided low dimension
Theorem 1.2 assumes only that P P P F F F κ \kappa κ Q Q Q Y Y Y y ∈ Y y \in Y y ∈ Y proxy μ ( y ) ∈ Y \mu(y) \in Y μ ( y ) ∈ Y Q Q Q Y Y Y 1 + 3 η 1+3\eta 1 + 3 η Q ∪ μ ( Y ) Q \cup\mu(Y) Q ∪ μ ( Y ) O ( ( 32 / η ) κ log 2 ( 2 / η ) ) O((32/\eta)^\kappa\log^2(2/\eta)) O (( 32/ η ) κ log 2 ( 2/ η )) Y Y Y r r r Q Q Q q q q Q Q Q 8 r 8r 8 r q q q q q q Figure 4 ). Every point of Q Q Q Q Q Q 6 r 6r 6 r Q Q Q Y Y Y ( 1 + 3 η ) 2 z (1+3\eta)^{2z} ( 1 + 3 η ) 2 z
Figure 4. The compression of Theorem 9.1 (not to scale). The points y y y y ′ y' y ′ r r r Q Q Q q q q η r \eta r η r x y x_y x y Q Q Q y y y dist ( y , q ) < 3 r \operatorname{dist}(y,q)<3r dist ( y , q ) < 3 r 8 r 8r 8 r q q q Q Q Q q q q y y y y ′ y' y ′ Q Q Q 1 + 3 η 1+3\eta 1 + 3 η
Euclidean space
In Euclidean space, centers may be placed anywhere, and the triangle bound ignores the shape of a block. Take the representative a i a_i a i i i i Ψ P i ( c ) = ∑ p ∈ P i ∥ p − c ∥ z \Psi_{P_i}(c)=\sum_{p\in P_i}\lVert p-c\rVert^z Ψ P i ( c ) = ∑ p ∈ P i ∥ p − c ∥ z Ψ P i ( a i ) = A i \Psi_{P_i}(a_i)=A_i Ψ P i ( a i ) = A i k k k a i a_i a i P i P_i P i y 1 , … , y w i y_1,\ldots,y_{w_i} y 1 , … , y w i
∥ p − y ∥ 2 = ∥ p − a i ∥ 2 + 2 ⟨ p − a i , a i − y ⟩ + ∥ a i − y ∥ 2 . \lVert p-y\rVert^2=\lVert p-a_i\rVert^2+2\langle p-a_i,a_i-y\rangle+\lVert a_i-y\rVert^2. ∥ p − y ∥ 2 = ∥ p − a i ∥ 2 + 2 ⟨ p − a i , a i − y ⟩ + ∥ a i − y ∥ 2 . The expectation of the middle term, summed over p ∈ P i p\in P_i p ∈ P i 2 w i ⟨ ∑ p ∈ P i ( p − a i ) , ∑ r ( a i − y r ) ⟩ = 0 \frac{2}{w_i}\left\langle\sum_{p\in P_i}(p-a_i),\sum_r(a_i-y_r)\right\rangle=0 w i 2 ⟨ ∑ p ∈ P i ( p − a i ) , ∑ r ( a i − y r ) ⟩ = 0 ( 1 , 1 ) (1,1) ( 1 , 1 ) β = 5 \beta=5 β = 5 3 2 = 9 3^2=9 3 2 = 9
Certificates. For z ≥ 1 z\geq1 z ≥ 1 ( u , g , v ) ∈ R 3 (u,g,v)\in\mathbb{R}^3 ( u , g , v ) ∈ R 3 z z z
∣ 1 + t ∣ z ≤ u + g t + v ∣ t ∣ z for all t ∈ R . (5) \lvert1+t\rvert^z\leq u+gt+v\lvert t\rvert^z\quad\text{for all }t\in\mathbb{R}. \tag*{(5)} ∣ 1 + t ∣ z ≤ u + g t + v ∣ t ∣ z for all t ∈ R . (5) For z > 1 z>1 z > 1 p , a , y ∈ R d p,a,y\in\mathbb{R}^d p , a , y ∈ R d
∥ p − y ∥ z ≤ u ∥ p − a ∥ z + g ∥ p − a ∥ z − 2 ⟨ p − a , a − y ⟩ + v ∥ a − y ∥ z . \lVert p-y\rVert^z\leq u\lVert p-a\rVert^z+g\lVert p-a\rVert^{z-2}\langle p-a,a-y\rangle+v\lVert a-y\rVert^z. ∥ p − y ∥ z ≤ u ∥ p − a ∥ z + g ∥ p − a ∥ z − 2 ⟨ p − a , a − y ⟩ + v ∥ a − y ∥ z . The proof maximizes over the angle between p − a p-a p − a a − y a-y a − y P i P_i P i
g w i ⟨ ∑ p ∈ P i ∥ p − a i ∥ z − 2 ( p − a i ) , ∑ r = 1 w i ( a i − y r ) ⟩ , \frac{g}{w_i}\left\langle\sum_{p\in P_i}\lVert p-a_i\rVert^{z-2}(p-a_i),\sum_{r=1}^{w_i}(a_i-y_r)\right\rangle, w i g ⟨ p ∈ P i ∑ ∥ p − a i ∥ z − 2 ( p − a i ) , r = 1 ∑ w i ( a i − y r ) ⟩ , and the first sum is − 1 z ∇ Ψ P i ( a i ) = 0 -\frac{1}{z}\nabla\Psi_{P_i}(a_i)=0 − z 1 ∇ Ψ P i ( a i ) = 0 z > 1 z>1 z > 1 ( u , v ) (u,v) ( u , v ) B z B_z B z Theorem 1.1 . The triangle bound corresponds to the certificate ( 3 z − 1 , 0 , ( 3 / 2 ) z − 1 ) (3z^{-1},0,(3/2)^{z-1}) ( 3 z − 1 , 0 , ( 3/2 ) z − 1 ) z ≥ 1 z\geq1 z ≥ 1 g g g B z B_z B z
Proposition 2.7 (The constant B z B_z B z
(a) B z = sup E ∣ 1 + ξ ∣ z , where the supremum is over all real random variables ξ with finite support, E [ ξ ] = 0 and E ∣ ξ ∣ z = 2 z . \text{(a)}\quad B_z=\sup\mathbb{E}|1+\xi|^z,
\quad\text{where the supremum is over all real random variables }\xi\text{ with finite support,}
\quad\mathbb{E}[\xi]=0\text{ and }\mathbb{E}|\xi|^z=2^z. (a) B z = sup E ∣1 + ξ ∣ z , where the supremum is over all real random variables ξ with finite support, E [ ξ ] = 0 and E ∣ ξ ∣ z = 2 z . (b) B z < 3 z for every z > 1. Moreover, B 1 = 3 , B 2 = 5 , B 3 / 2 ≈ 4.1397 , B 3 ≈ 16.8308 and B 4 ≈ 53.4056. \text{(b)}\quad B_z<3^z\text{ for every }z>1.
\quad\text{Moreover, }B_1=3,\ B_2=5,\ B_{3/2}\approx4.1397,\ B_3\approx16.8308\text{ and }B_4\approx53.4056. (b) B z < 3 z for every z > 1. Moreover, B 1 = 3 , B 2 = 5 , B 3/2 ≈ 4.1397 , B 3 ≈ 16.8308 and B 4 ≈ 53.4056. (c) For all u ≥ 1 and v ≥ 0 with u + 2 z v < B z , there is a finite set Q ⊂ R such that the block Q \text{(c)}\quad\text{For all }u\ge1\text{ and }v\ge0\text{ with }u+2^zv<B_z,\text{ there is a finite set }Q\subset\mathbb{R}\text{ such that the block }Q (c) For all u ≥ 1 and v ≥ 0 with u + 2 z v < B z , there is a finite set Q ⊂ R such that the block Q with an optimal center of Q as its representative is not ( u , v ) -movable. \text{with an optimal center of }Q\text{ as its representative is not }(u,v)\text{-movable.} with an optimal center of Q as its representative is not ( u , v ) -movable. For z = 1 z=1 z = 1 B 1 = 3 B_1=3 B 1 = 3
Candidates and nearly optimal centers. The framework needs a finite set of locations, and d d d ϑ ∈ ( 0 , 1 ] \vartheta\in(0,1] ϑ ∈ ( 0 , 1 ] Ψ \Psi Ψ O ( n 2 log n ) O(n^2\log n) O ( n 2 log n ) O ( z z 2 16 z / ϑ 2 ) O(z^{z^2 16^{z}/\vartheta^2}) O ( z z 2 1 6 z / ϑ 2 ) Q Q Q Q Q Q Q Q Q 1 + ϑ 1+\vartheta 1 + ϑ P P P ( u + z − 1 z E , v + 1 z E ) (u+\frac{z-1}{z}E,v+\frac{1}{z}E) ( u + z z − 1 E , v + z 1 E ) E E E 0 0 0 ϑ \vartheta ϑ d d d
Hard capacities and general metrics
For a finite family S S S F F F cost ( S ) \operatorname{cost}(S) cost ( S ) S S S U U U F F F cost ( F ) ≤ OPT \operatorname{cost}(F)\le\operatorname{OPT} cost ( F ) ≤ OPT projection lemma (Theorem 11.1) takes such a family S S S S ∗ S^* S ∗ ∣ S ∗ ∣ |S^*| ∣ S ∗ ∣ S S S ( cost ( S ) 1 / z + 2 cost ( S ∗ ) 1 / z ) z (\operatorname{cost}(S)^{1/z}+2\operatorname{cost}(S^*)^{1/z})^z ( cost ( S ) 1/ z + 2 cost ( S ∗ ) 1/ z ) z k k k S = F S=F S = F [74 ] . The clients induce a transport plan from the centers of S ∗ S^* S ∗ S S S S S S ∣ S ∗ ∣ |S^*| ∣ S ∗ ∣ S S S S = F S=F S = F Theorems 1.1 and 1.2 as S ∗ S^* S ∗ Theorem 1.3 .
In general metrics, Theorem 1.4 starts from one center per block and, while more than K K K A ≤ ρ OPT A\le\rho\operatorname{OPT} A ≤ ρ OPT A A A S = F S=F S = F S ∗ S^* S ∗ 2 k − 1 2k-1 2 k − 1 ( cost ( F ) 1 / z + 2 A 1 / z ) z ≤ ( 1 + 2 ρ 1 / z ) z OPT (\operatorname{cost}(F)^{1/z}+2A^{1/z})^z\le(1+2\rho^{1/z})^z\operatorname{OPT} ( cost ( F ) 1/ z + 2 A 1/ z ) z ≤ ( 1 + 2 ρ 1/ z ) z OPT γ = 1 + 2 ⋅ 3 \gamma=1+2\cdot3 γ = 1 + 2 ⋅ 3 k k k ρ = 3 + δ \rho=3+\delta ρ = 3 + δ k + r k+r k + r ζ OPT \zeta\operatorname{OPT} ζ OPT S ∗ S^* S ∗ k k k ( ζ 1 / z + 2 ) z OPT (\zeta^{1/z}+2)^z\operatorname{OPT} ( ζ 1/ z + 2 ) z OPT ( ( ζ 1 / z + 2 ) z − ζ ) OPT / r ((\zeta^{1/z}+2)^z-\zeta)\operatorname{OPT}/r (( ζ 1/ z + 2 ) z − ζ ) OPT / r [46 , 8 ] . For k k k 2 OPT / r 2\operatorname{OPT}/r 2 OPT / r K K K 2 OPT ∑ r = ⌈ ε k ⌉ + 1 k − 1 1 / r ≈ 2 ln ( 1 / ε ) OPT 2\operatorname{OPT}\sum_{r=\lceil\varepsilon k\rceil+1}^{k-1}1/r\approx2\ln(1/\varepsilon)\operatorname{OPT} 2 OPT ∑ r = ⌈ ε k ⌉ + 1 k − 1 1/ r ≈ 2 ln ( 1/ ε ) OPT z z z y ′ = ( y 1 / z + 2 ) z − y y'=(y^{1/z}+2)^z-y y ′ = ( y 1/ z + 2 ) z − y ∑ r 1 / r ≈ ln ( 1 / ε ) \sum_r1/r\approx\ln(1/\varepsilon) ∑ r 1/ r ≈ ln ( 1/ ε ) L z ( ρ , ε ) L_z(\rho,\varepsilon) L z ( ρ , ε )
Related Work
Capacitated clustering in general metrics. The best known polynomial-time ratio for capacitated k k k k k k O ( log k ) O(\log k) O ( log k ) [3 ] . For uniform capacities, the reverse greedy algorithm, which deletes centers one at a time, has ratio O ( log n ) O(\log n) O ( log n ) [8 ] ; its analysis builds on that of Chrobak, Kenyon and Young [46 ] for uncapacitated k k k Theorem 1.4 runs it from the blocks instead of from all locations. Constant ratios are known if the capacities may be exceeded by a constant factor [44 , 47 BFRS15, 73 ] or by a factor of 1 + ε 1+\varepsilon 1 + ε 1 + δ 1+\delta 1 + δ ( 5 + 5 / δ ) k (5+5/\delta)k ( 5 + 5/ δ ) k [70 ] , and ratio 7 + δ 7+\delta 7 + δ 2 k − 1 2k-1 2 k − 1 [16 ] . With ( 1 + ε ) k (1+\varepsilon)k ( 1 + ε ) k exp ( O ( 1 / ε 2 ) ) \exp(O(1/\varepsilon^2)) exp ( O ( 1/ ε 2 )) [74 ] and O ( ε − 2 log ( 1 / ε ) ) O(\varepsilon^{-2}\log(1/\varepsilon)) O ( ε − 2 log ( 1/ ε )) [75 ] , and Dai and Li [DL27] recently gave ratio O ( ε − 5 ) O(\varepsilon^{-5}) O ( ε − 5 ) n n n 1 / ε 1/\varepsilon 1/ ε [74 ] , so the ratio of [75 ] carries over to hard capacities. The natural LP relaxation has an unbounded integrality gap even if the capacities or the number of centers may be exceeded by a factor of 2 − c 2-c 2 − c c > 0 c>0 c > 0 [74 BFRS15] . For capacitated k k k [65 ] gave a constant ratio in R d \mathbb{R}^d R d k k k 2 + δ 2+\delta 2 + δ [34 ] , and local search gives ratio 3 + δ 3+\delta 3 + δ [9 ] ; k k k 9 + δ 9+\delta 9 + δ [13 ] , improved to 3 + 2 2 + δ 3+2\sqrt{2}+\delta 3 + 2 2 + δ [42 ] and 4.9 + δ 4.9+\delta 4.9 + δ z z z [62 ] .
Approximation schemes in low dimensions. Without capacities, approximation schemes in low dimensions use dynamic programming over hierarchical decompositions [14 , 71 , 90 , 32 ] or local search [35 , 57 ] . They are also known in planar and minor-free graphs [35 CAPP19b] , in graphs of bounded highway dimension [FFKP18, FS21] and, in time f ( k , δ ) ⋅ n O ( 1 ) f(k,\delta)\cdot n^{O(1)} f ( k , δ ) ⋅ n O ( 1 ) [1 ] ; for k k k [30 , 32 ] . With capacities, a dynamic program must count the clients that enter or leave every cell, and local search can be arbitrarily bad [31 ] . Arora, Raghavan and Rao [14 ] gave a quasi-polynomial-time scheme for capacitated k k k 1 + ε 1+\varepsilon 1 + ε [31 ] gave schemes that respect uniform capacities, in quasi-polynomial time for k k k k k k k k k [32 ] do without capacities. It uses ( log n / δ ) O ( κ ) (\log n/\delta)^{O(\kappa)} ( log n / δ ) O ( κ ) κ \kappa κ ( 1 + O ( ε ) ) k (1+O(\varepsilon))k ( 1 + O ( ε )) k 3 k 3k 3 k § 2.5 ; so a constant number of portals per cell suffices. For planar and minor-free metrics, the quasi-polynomial-time schemes for capacitated k k k 1 + δ 1+\delta 1 + δ f ( k , δ ) ⋅ n O ( 1 ) f(k,\delta)\cdot n^{O(1)} f ( k , δ ) ⋅ n O ( 1 ) [22 ] ; see [DGJ25, JK23] for capacitated clustering with outliers.
Extra centers. Opening more than k k k k k k 2 ( 1 + 1 / ε ) 2(1+1/\varepsilon) 2 ( 1 + 1/ ε ) ( 1 + ε ) k (1+\varepsilon)k ( 1 + ε ) k [78 ] , and local search gives ratio 1 + δ 1 + \delta 1 + δ ( 3 + 5 / δ ) k (3 + 5/\delta)k ( 3 + 5/ δ ) k [70 ] ; for k k k [82 ] decreases from 9 + δ 9 + \delta 9 + δ k k k 2.59 2.59 2.59 2 k 2k 2 k [79 ] achieved ratio 1 + δ 1 + \delta 1 + δ O ( ( 1 + 1 / δ ) k log n ) O((1 + 1/\delta)k \log n) O (( 1 + 1/ δ ) k log n ) k k k O ( k ) O(k) O ( k ) O ( k log k ) O(k \log k) O ( k log k ) [7 , 11 , 91 , 84 ] . Li and Svensson [76 ] showed that for k k k k + O ( 1 ) k + O(1) k + O ( 1 ) k k k ( 1 + ε ) k (1 + \varepsilon)k ( 1 + ε ) k k k k [38 ] and for k k k [29 ] , and ( 1 + ε ) k (1 + \varepsilon)k ( 1 + ε ) k k k k k k k [56 ] and for k k k f ( ε ) ⋅ n O ( 1 ) f(\varepsilon) \cdot n^{O(1)} f ( ε ) ⋅ n O ( 1 ) [4 , 86 ] .
Parameterized algorithms and coresets. Capacitated k k k W [ 2 ] W[2] W [ 2 ] k k k [3 ] . In time f ( k , δ ) ⋅ n O ( 1 ) f(k,\delta) \cdot n^{O(1)} f ( k , δ ) ⋅ n O ( 1 ) 7 + δ 7 + \delta 7 + δ [3 ] and 3 + δ 3 + \delta 3 + δ k k k 9 + δ 9 + \delta 9 + δ [52 ] obtained a tight ratio 3 + δ 3 + \delta 3 + δ [54 , 26 , 60 , 59 , 24 ] . Under the gap exponential time hypothesis, no algorithm with such a running time achieves a ratio better than 1 + 2 / e 1 + 2/e 1 + 2/ e k k k 1 + 8 / e 1 + 8/e 1 + 8/ e k k k [33 ] . A coreset is a small weighted set of points on which the cost of every solution is preserved up to a factor of 1 ± δ 1 \pm\delta 1 ± δ k k k 1 / δ 1/\delta 1/ δ log n \log n log n k k k n n n [17 , 63 ] ; see also [19 ] .
Related problems. Capacitated k k k [25 , 72 ] and for non-uniform capacities [45 , 2 ] , with ratio 9 9 9 [2 ] , and it is NP-hard to approximate within 3 − δ 3 - \delta 3 − δ [45 ] . In capacitated facility location, every center has an opening cost, and the number of centers is not bounded. Local search gives constant ratios [70 , 48 , 88 ZCY05] , such as 3 3 3 [12 ] and 5 5 5 [21 ] ; LP-based algorithms are also known [77 , 15 ] , and with soft capacities, ratio 2 2 2 [87 ] . Capacitated vehicle routing admits quasi-polynomial-time approximation schemes in the Euclidean plane [53 ] and in graphs of bounded treewidth, doubling dimension or highway dimension [69 ] .
Applications. Capacitated and balanced clustering is used for political districting [64 , 36 ] ; for the k k k [10 ] . It is also used for point distributions in computer graphics [28 ] and for balanced clustering in data mining [20 , 81 ] .
Preliminaries
This section collects standard facts used in the proofs; the setting is defined at the beginning of § 2 . As usual, arithmetic operations and evaluations of dist ( x , y ) 2 \operatorname{dist}(x,y)^2 dist ( x , y ) 2
Doubling dimension. For a metric space ( X , dist ) (X,\operatorname{dist}) ( X , dist ) x ∈ X x \in X x ∈ X r ≥ 0 r \ge0 r ≥ 0 B ( x , r ) = { y ∈ X : dist ( x , y ) ≤ r } B(x,r) = \{y \in X : \operatorname{dist}(x,y) \le r\} B ( x , r ) = { y ∈ X : dist ( x , y ) ≤ r } X X X k k k B ( x , r ) B(x,r) B ( x , r ) 2 κ 2^\kappa 2 κ r / 2 r/2 r /2 X X X R d \mathbb{R}^d R d 3 d 3d 3 d
Fact 4.1. Let X X X κ \kappa κ x ∈ X x \in X x ∈ X 0 < s ≤ r 0 < s \le r 0 < s ≤ r B ( x , r ) B(x,r) B ( x , r ) s s s ( 4 r / s ) κ (4r/s)^\kappa ( 4 r / s ) κ
Proof. Applying the definition j = ⌈ log 2 ( 2 r / s ) ⌉ j=\lceil\log_2(2r/s)\rceil j = ⌈ log 2 ( 2 r / s )⌉ B ( x , r ) B(x,r) B ( x , r ) 2 j κ < ( 4 r / s ) κ 2^{j\kappa} < (4r/s)^\kappa 2 j κ < ( 4 r / s ) κ 2 − j r ≤ s / 2 2^{-j}r \le s/2 2 − j r ≤ s /2
Minkowski’s inequality. The triangle inequality and Minkowski’s inequality ( ∑ t ( a t + b t ) z ) 1 / z ≤ ( ∑ t a t z ) 1 / z + ( ∑ t b t z ) 1 / z \left(\sum_t(a_t+b_t)^z\right)^{1/z} \le\left(\sum_t a_t^z\right)^{1/z}+\left(\sum_t b_t^z\right)^{1/z} ( ∑ t ( a t + b t ) z ) 1/ z ≤ ( ∑ t a t z ) 1/ z + ( ∑ t b t z ) 1/ z ( x t , y t , p t ) (x_t,y_t,p_t) ( x t , y t , p t )
( ∑ t dist ( x t , y t ) z ) 1 / z ≤ ( ∑ t dist ( x t , p t ) z ) 1 / z + ( ∑ t dist ( p t , y t ) z ) 1 / z . (6) \left(\sum_t \operatorname{dist}(x_t,y_t)^z\right)^{1/z} \le\left(\sum_t \operatorname{dist}(x_t,p_t)^z\right)^{1/z}+\left(\sum_t \operatorname{dist}(p_t,y_t)^z\right)^{1/z}. \tag*{(6)} ( t ∑ dist ( x t , y t ) z ) 1/ z ≤ ( t ∑ dist ( x t , p t ) z ) 1/ z + ( t ∑ dist ( p t , y t ) z ) 1/ z . (6) By the convexity of t ↦ t z t \mapsto t^z t ↦ t z ( a 1 / z + b 1 / z ) z ≤ 2 z − 1 ( a + b ) (a^{1/z}+b^{1/z})^z \le2^{z-1}(a+b) ( a 1/ z + b 1/ z ) z ≤ 2 z − 1 ( a + b ) a , b ≥ 0 a,b \ge0 a , b ≥ 0
Transport plans. Let I I I J J J w i w_i w i ( i ∈ I ) (i \in I) ( i ∈ I ) ℓ j \ell_j ℓ j ( j ∈ J ) (j \in J) ( j ∈ J ) ∑ i w i = ∑ j ℓ j \sum_i w_i=\sum_j \ell_j ∑ i w i = ∑ j ℓ j π ∈ Z ≥ 0 I × J \pi\in\mathbb{Z}_{\ge0}^{I\times J} π ∈ Z ≥ 0 I × J w i w_i w i ℓ j \ell_j ℓ j supp ( π ) \operatorname{supp}(\pi) supp ( π ) I ∪ J I \cup J I ∪ J i j ij ij π i j > 0 \pi_{ij}>0 π ij > 0 i j ij ij π \pi π π i j \pi_{ij} π ij i j ij ij
Fact 4.2 (Cycle cancelling). Every integral transport plan π \pi π supp ( π ) \operatorname{supp}(\pi) supp ( π )
Proof. A cycle in the support has even length. Adding 1 to every other entry on the cycle and subtracting 1 from the others keeps all row and column sums, and the two directions change the cost by opposite amounts. Moving in a direction that does not increase the cost, by the largest amount that keeps the entries nonnegative, removes an edge of the cycle and adds none.
Uncapacitated clustering. We use the following algorithm to compute initial solutions.
Theorem 4.3 ([57 ] ). For fixed κ ≥ 1 \kappa\ge1 κ ≥ 1 z ≥ 1 z \ge1 z ≥ 1 η > 0 \eta> 0 η > 0 P P P F F F P ∪ F P \cup F P ∪ F κ \kappa κ k k k F F F ( 1 + η ) (1+\eta) ( 1 + η )
The algorithm is local search with swaps of κ O ( κ ) ( 2 z / η ) O ( 2 z κ / η ) \kappa^{O(\kappa)}(2^z/\eta)^{O(2^z\kappa/\eta)} κ O ( κ ) ( 2 z / η ) O ( 2 z κ / η ) k k k z ≥ 1 z \ge1 z ≥ 1 [57 ] , and it allows several clients at the same point.
Proof of the Rescue Lemma
The proof of Theorem 2.5 uses a standard separator for forests.
Lemma 5.1 (Forest separator). Let G G G L ≥ 1 L \ge1 L ≥ 1 Z Z Z ∣ V ( G ) ∣ / ( L + 1 ) |V(G)|/(L+1) ∣ V ( G ) ∣/ ( L + 1 ) G − Z G-Z G − Z L L L Z Z Z
Proof. Root every tree of G G G x x x x x x Z Z Z x x x L L L x x x Z Z Z G − Z G-Z G − Z x x x x x x L L L x x x Z Z Z L + 1 L+1 L + 1 x x x Z Z Z x x x x x x ∣ Z ∣ ( L + 1 ) ≤ ∣ V ( G ) ∣ |Z|(L+1) \leq|V(G)| ∣ Z ∣ ( L + 1 ) ≤ ∣ V ( G ) ∣
Proof of Theorem 2.5. The forest supp ( π ) \operatorname{supp}(\pi) supp ( π ) ∣ I ∣ + ∣ J ∣ ≤ 3 k − 1 |I|+|J|\leq3k-1 ∣ I ∣ + ∣ J ∣ ≤ 3 k − 1 Z Z Z ∣ Z ∣ < 3 k / ( L + 1 ) |Z|<3k/(L+1) ∣ Z ∣ < 3 k / ( L + 1 ) supp ( π ) − Z \operatorname{supp}(\pi)-Z supp ( π ) − Z L L L Z I = Z ∩ I Z_I=Z\cap I Z I = Z ∩ I Z J = Z ∩ J Z_J=Z\cap J Z J = Z ∩ J Y Y Y supp ( π ) − Z \operatorname{supp}(\pi)-Z supp ( π ) − Z I Y = I ∩ Y I_Y=I\cap Y I Y = I ∩ Y J Y = J ∩ Y J_Y=J\cap Y J Y = J ∩ Y d Y = ∣ I Y ∣ − ∣ J Y ∣ d_Y=|I_Y|-|J_Y| d Y = ∣ I Y ∣ − ∣ J Y ∣ W Y = ∑ i ∈ I Y W i W_Y=\sum_{i\in I_Y}W_i W Y = ∑ i ∈ I Y W i
For a set Y \mathcal{Y} Y R = Z I ∪ ⋃ Y ∈ Y I Y R=Z_I\cup\bigcup_{Y\in\mathcal{Y}}I_Y R = Z I ∪ ⋃ Y ∈ Y I Y J 0 = ⋃ Y ∈ Y J Y J_0=\bigcup_{Y\in\mathcal{Y}}J_Y J 0 = ⋃ Y ∈ Y J Y π i j > 0 \pi_{ij}>0 π ij > 0 j ∈ J Y j\in J_Y j ∈ J Y i i i Z I Z_I Z I Y Y Y
∣ R ∣ + ∣ J ∣ − ∣ J 0 ∣ = ∣ J ∣ + ∣ Z I ∣ + ∑ Y ∈ Y d Y . (7) |R|+|J|-|J_0|=|J|+|Z_I|+\sum_{Y\in\mathcal{Y}}d_Y. \tag*{(7)} ∣ R ∣ + ∣ J ∣ − ∣ J 0 ∣ = ∣ J ∣ + ∣ Z I ∣ + Y ∈ Y ∑ d Y . (7) If b < ∣ Z I ∣ + L b<|Z_I|+L b < ∣ Z I ∣ + L R = J 0 = ∅ R=J_0=\varnothing R = J 0 = ∅ ( b − L ) / k < ∣ Z I ∣ / k < 3 / ( L + 1 ) (b-L)/k<|Z_I|/k<3/(L+1) ( b − L ) / k < ∣ Z I ∣/ k < 3/ ( L + 1 )
Now let b ≥ ∣ Z I ∣ + L b\geq|Z_I|+L b ≥ ∣ Z I ∣ + L Y 0 \mathcal{Y}_0 Y 0 d Y ≤ 0 d_Y\leq0 d Y ≤ 0 1 1 1 L L L W + W_+ W + d + d_+ d + (7) , we may spend the budget
B = k + b − ∣ J ∣ − ∣ Z I ∣ − ∑ Y ∈ Y 0 d Y ≥ b − ∣ Z I ∣ ≥ L B=k+b-|J|-|Z_I|-\sum_{Y\in\mathcal{Y}_0}d_Y\geq b-|Z_I|\geq L B = k + b − ∣ J ∣ − ∣ Z I ∣ − Y ∈ Y 0 ∑ d Y ≥ b − ∣ Z I ∣ ≥ L on components of positive excess. We sort these components by W Y / d Y W_Y/d_Y W Y / d Y Y 1 \mathcal{Y}_1 Y 1 B B B Y = Y 0 ∪ Y 1 \mathcal{Y}=\mathcal{Y}_0\cup\mathcal{Y}_1 Y = Y 0 ∪ Y 1 (7) .
It remains to prove (iii). If Y 1 \mathcal{Y}_1 Y 1 R = I R=I R = I Y 1 \mathcal{Y}_1 Y 1 L L L Y 1 \mathcal{Y}_1 Y 1 B − L ≥ 0 B-L\geq0 B − L ≥ 0 Y 1 \mathcal{Y}_1 Y 1 ∑ Y ∈ Y 1 W Y ≥ B − L d + W + \sum_{Y\in\mathcal{Y}_1}W_Y\geq\frac{B-L}{d_+}W_+ ∑ Y ∈ Y 1 W Y ≥ d + B − L W + ∑ Y d Y = ( ∣ I ∣ − ∣ Z I ∣ ) − ( ∣ J ∣ − ∣ Z J ∣ ) \sum_Yd_Y=(|I|-|Z_I|)-(|J|-|Z_J|) ∑ Y d Y = ( ∣ I ∣ − ∣ Z I ∣ ) − ( ∣ J ∣ − ∣ Z J ∣ ) ∣ I ∣ ≤ 2 k − 1 |I|\leq2k-1 ∣ I ∣ ≤ 2 k − 1
d + = ∑ Y d Y − ∑ Y ∈ Y 0 d Y < B + ( k − b ) + ∣ Z J ∣ . d_+=\sum_Yd_Y-\sum_{Y\in\mathcal{Y}_0}d_Y<B+(k-b)+|Z_J|. d + = Y ∑ d Y − Y ∈ Y 0 ∑ d Y < B + ( k − b ) + ∣ Z J ∣. The function t ↦ ( t − L ) / ( t + k − b + ∣ Z J ∣ ) t\mapsto(t-L)/(t+k-b+|Z_J|) t ↦ ( t − L ) / ( t + k − b + ∣ Z J ∣ ) t > 0 t>0 t > 0 B ≥ b − ∣ Z I ∣ B\geq b-|Z_I| B ≥ b − ∣ Z I ∣
B − L d + ≥ B − L B + k − b + ∣ Z J ∣ ≥ b − ∣ Z I ∣ − L k − ∣ Z I ∣ + ∣ Z J ∣ ≥ b − ∣ Z ∣ − L k . \frac{B-L}{d_+}\geq\frac{B-L}{B+k-b+|Z_J|}\geq\frac{b-|Z_I|-L}{k-|Z_I|+|Z_J|}\geq\frac{b-|Z|-L}{k}. d + B − L ≥ B + k − b + ∣ Z J ∣ B − L ≥ k − ∣ Z I ∣ + ∣ Z J ∣ b − ∣ Z I ∣ − L ≥ k b − ∣ Z ∣ − L . The last inequality says that the separator and the rounding lose at most ( ∣ Z ∣ + L ) / k (|Z|+L)/k ( ∣ Z ∣ + L ) / k b / k b/k b / k § 2.3 : the blocks of Z I Z_I Z I ∣ Z I ∣ |Z_I| ∣ Z I ∣ Z J Z_J Z J ∣ Z J ∣ |Z_J| ∣ Z J ∣ ( b − ∣ Z I ∣ − L ) ∣ Z I ∣ + ∣ Z J ∣ ( k − b + L + ∣ Z J ∣ ) ≥ 0 (b-|Z_I|-L)|Z_I|+|Z_J|(k-b+L+|Z_J|)\geq0 ( b − ∣ Z I ∣ − L ) ∣ Z I ∣ + ∣ Z J ∣ ( k − b + L + ∣ Z J ∣ ) ≥ 0 b ≥ ∣ Z I ∣ + L b\geq|Z_I|+L b ≥ ∣ Z I ∣ + L b ≤ k b\leq k b ≤ k R R R Z I Z_I Z I Y 0 \mathcal{Y}_0 Y 0 ( b − ∣ Z ∣ − L ) / k ≤ 1 (b-|Z|-L)/k\leq1 ( b − ∣ Z ∣ − L ) / k ≤ 1
∑ i ∈ R W i ≥ ∑ i ∈ Z I W i + ∑ Y ∈ Y 0 W Y + b − ∣ Z ∣ − L k W + ≥ b − ∣ Z ∣ − L k ∑ i ∈ I W i , \sum_{i\in R}W_i\geq\sum_{i\in Z_I}W_i+\sum_{Y\in\mathcal{Y}_0}W_Y+\frac{b-|Z|-L}{k}W_+\geq\frac{b-|Z|-L}{k}\sum_{i\in I}W_i, i ∈ R ∑ W i ≥ i ∈ Z I ∑ W i + Y ∈ Y 0 ∑ W Y + k b − ∣ Z ∣ − L W + ≥ k b − ∣ Z ∣ − L i ∈ I ∑ W i , and ∣ Z ∣ < 3 k / ( L + 1 ) |Z|<3k/(L+1) ∣ Z ∣ < 3 k / ( L + 1 )
Hierarchical Decompositions
This section constructs the decompositions on which the network of § 7 is built. The network routes the transport plan of Theorem 2.4 accurately along a pair of points unless the decomposition cuts the pair at a level much higher than its distance. We therefore need a family of decompositions such that, for every list of pairs, some member of the family cuts few pairs of the list badly. Randomized decompositions with this property are standard for doubling metrics [61 , 90 , 32 ] . We give a polynomial-size family in which all levels share one hash function and one radius parameter, so that the family can be enumerated.
Throughout this section, X X X κ \kappa κ X X X 2 2 2 H H H 2 H ≥ diam ( X ) 2^H \ge\operatorname{diam}(X) 2 H ≥ diam ( X ) H = O ( log Δ ( X ) ) H = O(\log\Delta(X)) H = O ( log Δ ( X ))
Definition 6.1 (Hierarchical decomposition). A hierarchical decomposition D \mathcal{D} D X X X P 0 , … , P H \mathcal{P}_0,\ldots,\mathcal{P}_H P 0 , … , P H X X X P H = { X } \mathcal{P}_H = \{X\} P H = { X } P h \mathcal{P}_h P h 2 h 2^h 2 h P h \mathcal{P}_h P h P h + 1 \mathcal{P}_{h+1} P h + 1 h < H h < H h < H P h \mathcal{P}_h P h cells of level h h h parent of a cell of level h < H h < H h < H h + 1 h+1 h + 1 x , y ∈ X x,y \in X x , y ∈ X cut level lev ( x , y ) \operatorname{lev}(x,y) lev ( x , y ) h h h x x x y y y h h h Λ ≥ 1 \Lambda\ge1 Λ ≥ 1 ( x , y ) (x,y) ( x , y ) Λ \Lambda Λ bad if 2 lev ( x , y ) > Λ ⋅ dist ( x , y ) 2^{\operatorname{lev}(x,y)} > \Lambda\cdot\operatorname{dist}(x,y) 2 lev ( x , y ) > Λ ⋅ dist ( x , y ) ( x , x ) (x,x) ( x , x )
Since the minimum distance is 2 2 2 0 0 0
Lemma 6.2 (Decomposition family). There is a family H \mathcal{H} H O ( ∣ X ∣ κ ) O(|X|^\kappa) O ( ∣ X ∣ κ ) X X X ∣ X ∣ |X| ∣ X ∣ H H H H \mathcal{H} H Λ ≥ 1 \Lambda\ge1 Λ ≥ 1 x , y ∈ X x,y \in X x , y ∈ X
Pr D ∼ H [ ( x , y ) is Λ -bad in D ] ≤ 2 11 κ Λ . \Pr_{\mathcal{D}\sim\mathcal{H}}\left[(x,y)\text{ is }\Lambda\text{-bad in }\mathcal{D}\right] \le\frac{2^{11\kappa}}{\Lambda}. D ∼ H Pr [ ( x , y ) is Λ -bad in D ] ≤ Λ 2 11 κ . Corollary 6.3 . For every Λ ≥ 1 \Lambda\ge1 Λ ≥ 1 E E E X X X D ∈ H \mathcal{D} \in\mathcal{H} D ∈ H 2 11 κ ∣ E ∣ / Λ 2^{11\kappa}|E|/\Lambda 2 11 κ ∣ E ∣/Λ E E E Λ \Lambda Λ D \mathcal{D} D
Proof. By Theorem 6.2 and linearity of expectation, the expected number of Λ \Lambda Λ E E E 2 11 κ ∣ E ∣ / Λ 2^{11\kappa}|E|/\Lambda 2 11 κ ∣ E ∣/Λ H \mathcal{H} H
The construction. For every level h < H h < H h < H X h ⊆ X X_h \subseteq X X h ⊆ X 2 h − 3 2^{h-3} 2 h − 3 X X X 2 h − 3 2^{h-3} 2 h − 3 X h X_h X h q q q ∣ X ∣ ≤ q < 2 ∣ X ∣ |X| \le q < 2|X| ∣ X ∣ ≤ q < 2∣ X ∣ X X X F q \mathbb{F}_q F q 0 , … , q − 1 0,\ldots,q-1 0 , … , q − 1 g g g F q \mathbb{F}_q F q g g g x x x X X X ( g ( x ) , x ) (g(x),x) ( g ( x ) , x )
For t ∈ [ 1 / 4 , 1 / 2 ] t \in[1/4,1/2] t ∈ [ 1/4 , 1/2 ] h < H h < H h < H Q h ( g , t ) \mathcal{Q}_h(g,t) Q h ( g , t ) x ∈ X x \in X x ∈ X x ′ ∈ X h x' \in X_h x ′ ∈ X h g g g dist ( x , x ′ ) ≤ t 2 h \operatorname{dist}(x,x') \le t2^h dist ( x , x ′ ) ≤ t 2 h X h X_h X h t 2 h ≥ 2 h − 2 > 2 h − 3 t2^h \ge2^{h-2} > 2^{h-3} t 2 h ≥ 2 h − 2 > 2 h − 3 2 t 2 h ≤ 2 h 2t2^h \le2^h 2 t 2 h ≤ 2 h P H ( g , t ) = { X } \mathcal{P}_H(g,t) = \{X\} P H ( g , t ) = { X } h < H h < H h < H P h ( g , t ) \mathcal{P}_h(g,t) P h ( g , t ) Q h ( g , t ) , … , Q H − 1 ( g , t ) \mathcal{Q}_h(g,t),\ldots,\mathcal{Q}_{H-1}(g,t) Q h ( g , t ) , … , Q H − 1 ( g , t ) C h ∩ ⋯ ∩ C H − 1 C_h \cap\cdots\cap C_{H-1} C h ∩ ⋯ ∩ C H − 1 C l C_l C l Q l ( g , t ) \mathcal{Q}_l(g,t) Q l ( g , t ) D ( g , t ) \mathcal{D}(g,t) D ( g , t )
We choose g g g F q \mathbb{F}_q F q t t t [ 1 / 4 , 1 / 2 ] [1/4,1/2] [ 1/4 , 1/2 ] g g g D ( g , t ) \mathcal{D}(g,t) D ( g , t ) t t t dist ( x , x ′ ) ≤ t 2 h \operatorname{dist}(x,x') \le t2^h dist ( x , x ′ ) ≤ t 2 h
{ 1 / 4 , 1 / 2 } ∪ { dist ( x , x ′ ) / 2 h ∈ [ 1 / 4 , 1 / 2 ] : x ∈ X , x ′ ∈ X h , h < H } . \{1/4,1/2\} \cup\{\operatorname{dist}(x,x')/2^h \in[1/4,1/2] : x \in X,\ x' \in X_h,\ h < H\}. { 1/4 , 1/2 } ∪ { dist ( x , x ′ ) / 2 h ∈ [ 1/4 , 1/2 ] : x ∈ X , x ′ ∈ X h , h < H } . For fixed x x x x ′ x' x ′ dist ( x , x ′ ) / 2 h ∈ [ 1 / 4 , 1 / 2 ] \operatorname{dist}(x,x')/2^h \in[1/4,1/2] dist ( x , x ′ ) / 2 h ∈ [ 1/4 , 1/2 ] 2 dist ( x , x ′ ) ≤ 2 h ≤ 4 dist ( x , x ′ ) 2\operatorname{dist}(x,x') \le2^h \le4\operatorname{dist}(x,x') 2 dist ( x , x ′ ) ≤ 2 h ≤ 4 dist ( x , x ′ ) h h h 2 ∣ X ∣ 2 + 2 2|X|^2+2 2∣ X ∣ 2 + 2 H \mathcal{H} H D ( g , t ) \mathcal{D}(g,t) D ( g , t ) g g g t t t q 5 ⋅ ( 2 ∣ X ∣ 2 + 1 ) = O ( ∣ X ∣ 7 ) q^5 \cdot(2|X|^2+1)=O(|X|^7) q 5 ⋅ ( 2∣ X ∣ 2 + 1 ) = O ( ∣ X ∣ 7 ) D ( g , t ) \mathcal{D}(g,t) D ( g , t ) H \mathcal{H} H H \mathcal{H} H
The analysis. The proof of Theorem 6.2 has three steps. Theorem 6.4 shows that a fixed point is first in a set Q Q Q g g g O ( 1 / ∣ Q ∣ ) O(1/|Q|) O ( 1/∣ Q ∣ ) [66 ] ; Theorem 6.5 uses this to bound the probability that one level cuts a small ball; and the proof of Theorem 6.2 sums these probabilities over the levels.
Claim 6.4. For every nonempty Q ⊆ X Q \subseteq X Q ⊆ X x ∈ Q x \in Q x ∈ Q x x x Q Q Q g g g 20 / ∣ Q ∣ 20/|Q| 20/∣ Q ∣
Proof. Let s = ∣ Q ∣ ≥ 2 s=|Q| \ge2 s = ∣ Q ∣ ≥ 2 g g g F q \mathbb{F}_q F q g ( x ) g(x) g ( x ) χ \chi χ y ∈ Q ∖ { x } y \in Q \setminus\{x\} y ∈ Q ∖ { x } g ( y ) < g ( x ) g(y)<g(x) g ( y ) < g ( x ) x x x χ = 0 \chi=0 χ = 0 g ( x ) g(x) g ( x ) χ \chi χ s − 1 s-1 s − 1 g ( x ) / q g(x)/q g ( x ) / q E χ = ( s − 1 ) g ( x ) / q \mathbb{E}\chi=(s-1)g(x)/q E χ = ( s − 1 ) g ( x ) / q E [ ( χ − E χ ) 4 ] ≤ E χ + 3 ( E χ ) 2 \mathbb{E}[(\chi-\mathbb{E}\chi)^4]\le\mathbb{E}\chi+3(\mathbb{E}\chi)^2 E [( χ − E χ ) 4 ] ≤ E χ + 3 ( E χ ) 2 E χ ≥ 1 \mathbb{E}\chi\ge1 E χ ≥ 1
Pr [ χ = 0 ∣ g ( x ) ] ≤ Pr [ ∣ χ − E χ ∣ ≥ E χ ∣ g ( x ) ] ≤ E χ + 3 ( E χ ) 2 ( E χ ) 4 ≤ 4 ( E χ ) 2 . \Pr[\chi=0\mid g(x)]\le\Pr[|\chi-\mathbb{E}\chi|\ge\mathbb{E}\chi\mid g(x)]\le\frac{\mathbb{E}\chi+3(\mathbb{E}\chi)^2}{(\mathbb{E}\chi)^4}\le\frac{4}{(\mathbb{E}\chi)^2}. Pr [ χ = 0 ∣ g ( x )] ≤ Pr [ ∣ χ − E χ ∣ ≥ E χ ∣ g ( x )] ≤ ( E χ ) 4 E χ + 3 ( E χ ) 2 ≤ ( E χ ) 2 4 . Now average over the q q q g ( x ) g(x) g ( x ) E χ < 1 \mathbb{E}\chi<1 E χ < 1 g ( x ) < q / ( s − 1 ) g(x)<q/(s-1) g ( x ) < q / ( s − 1 ) ( q / ( s − 1 ) + 1 ) / q ≤ 2 / ( s − 1 ) (q/(s-1)+1)/q\le2/(s-1) ( q / ( s − 1 ) + 1 ) / q ≤ 2/ ( s − 1 ) q ≥ s q\ge s q ≥ s g ( x ) = j ≥ q / ( s − 1 ) g(x)=j\ge q/(s-1) g ( x ) = j ≥ q / ( s − 1 )
1 q ∑ j ≥ q / ( s − 1 ) 4 q 2 ( s − 1 ) 2 j 2 ≤ 4 q ( s − 1 ) 2 ⋅ 2 ( s − 1 ) q = 8 s − 1 , \frac{1}{q}\sum_{j\ge q/(s-1)}\frac{4q^2}{(s-1)^2j^2}
\le\frac{4q}{(s-1)^2}\cdot\frac{2(s-1)}{q}
=\frac{8}{s-1}, q 1 j ≥ q / ( s − 1 ) ∑ ( s − 1 ) 2 j 2 4 q 2 ≤ ( s − 1 ) 2 4 q ⋅ q 2 ( s − 1 ) = s − 1 8 , using ∑ j ≥ c j − 2 ≤ 2 / c \sum_{j\ge c}j^{-2}\le2/c ∑ j ≥ c j − 2 ≤ 2/ c c ≥ 1 c\ge1 c ≥ 1 10 / ( s − 1 ) ≤ 20 / s 10/(s-1)\le20/s 10/ ( s − 1 ) ≤ 20/ s
Claim 6.5. For all x ∈ X x\in X x ∈ X r > 0 r>0 r > 0 h < H h<H h < H Q h ( g , t ) Q_h(g,t) Q h ( g , t ) B ( x , r ) B(x,r) B ( x , r ) 2 9 κ r / 2 h 2^9\kappa r/2^h 2 9 κ r / 2 h
Proof. If r > 2 h − 3 r>2^{h-3} r > 2 h − 3 r ≤ 2 h − 3 r\le2^{h-3} r ≤ 2 h − 3 t t t X − = X h ∩ B ( x , t 2 h − r ) X_- = X_h \cap B(x,t2^h-r) X − = X h ∩ B ( x , t 2 h − r ) X + = X h ∩ B ( x , t 2 h + r ) X_+ = X_h \cap B(x,t2^h+r) X + = X h ∩ B ( x , t 2 h + r ) X − X_- X − t 2 h − r ≥ 2 h − 3 t2^h-r\ge2^{h-3} t 2 h − r ≥ 2 h − 3 X h X_h X h 2 h − 3 2^{h-3} 2 h − 3 x x x x ′ x' x ′ X + X_+ X + g g g X − X_- X − x ′ x' x ′ t 2 h t2^h t 2 h B ( x , r ) B(x,r) B ( x , r ) X h X_h X h t 2 h t2^h t 2 h B ( x , r ) B(x,r) B ( x , r ) X + X_+ X + x ′ x' x ′ B ( x , r ) B(x,r) B ( x , r ) x ′ x' x ′ X + ∖ X − X_+\setminus X_- X + ∖ X −
Pr g [ Q h ( g , t ) separates B ( x , r ) ] ≤ 20 ∣ X + ∖ X − ∣ ∣ X + ∣ ≤ 20 ln ∣ X + ∣ ∣ X − ∣ , \Pr_g[Q_h(g,t)\text{ separates }B(x,r)]\le20\frac{|X_+\setminus X_-|}{|X_+|}\le20\ln\frac{|X_+|}{|X_-|}, g Pr [ Q h ( g , t ) separates B ( x , r )] ≤ 20 ∣ X + ∣ ∣ X + ∖ X − ∣ ≤ 20 ln ∣ X − ∣ ∣ X + ∣ , using 1 − 1 / w ≤ ln w 1 - 1/w \le\ln w 1 − 1/ w ≤ ln w w ≥ 1 w \ge1 w ≥ 1 ψ ( s ) = ln ∣ X h ∩ B ( x , s ) ∣ \psi(s) = \ln|X_h \cap B(x,s)| ψ ( s ) = ln ∣ X h ∩ B ( x , s ) ∣ s ≥ 2 h − 3 s \ge2^{h-3} s ≥ 2 h − 3 X h ∩ B ( x , 5 ⋅ 2 h − 3 ) X_h \cap B(x,5 \cdot2^{h-3}) X h ∩ B ( x , 5 ⋅ 2 h − 3 ) 2 h − 3 2^{h-3} 2 h − 3 20 κ 20^\kappa 2 0 κ ψ ( 5 ⋅ 2 h − 3 ) ≤ κ ln 20 \psi(5 \cdot2^{h-3}) \le\kappa\ln20 ψ ( 5 ⋅ 2 h − 3 ) ≤ κ ln 20 t t t s = t 2 h s = t2^h s = t 2 h [ 2 h − 2 , 2 h − 1 ] [2^{h-2},2^{h-1}] [ 2 h − 2 , 2 h − 1 ]
E t [ ln ∣ X + ∣ ∣ X − ∣ ] = 4 2 h ∫ 2 h − 2 2 h − 1 ( ψ ( s + r ) − ψ ( s − r ) ) d s = 4 2 h ( ∫ 2 h − 1 − r 2 h − 1 + r ψ − ∫ 2 h − 2 − r 2 h − 2 + r ψ ) ≤ 4 2 h ⋅ 2 r ψ ( 5 ⋅ 2 h − 3 ) . \mathbb{E}_t\left[\ln\frac{|X^+|}{|X^-|}\right] = \frac{4}{2^h}\int_{2^{h-2}}^{2^{h-1}}\left(\psi(s+r)-\psi(s-r)\right)\,\mathrm{d}s = \frac{4}{2^h}\left(\int_{2^{h-1}-r}^{2^{h-1}+r}\psi-\int_{2^{h-2}-r}^{2^{h-2}+r}\psi\right) \le\frac{4}{2^h}\cdot2r\psi(5\cdot2^{h-3}). E t [ ln ∣ X − ∣ ∣ X + ∣ ] = 2 h 4 ∫ 2 h − 2 2 h − 1 ( ψ ( s + r ) − ψ ( s − r ) ) d s = 2 h 4 ( ∫ 2 h − 1 − r 2 h − 1 + r ψ − ∫ 2 h − 2 − r 2 h − 2 + r ψ ) ≤ 2 h 4 ⋅ 2 r ψ ( 5 ⋅ 2 h − 3 ) . Hence the probability is at most 20 ⋅ 8 κ ln 20 ⋅ r / 2 h ≤ 2 9 κ r / 2 h 20\cdot8\kappa\ln20\cdot r/2^h \le2^9\kappa r/2^h 20 ⋅ 8 κ ln 20 ⋅ r / 2 h ≤ 2 9 κ r / 2 h
Proof of Theorem 6.2. If no partition Q l ( g , t ) \mathcal{Q}_l(g,t) Q l ( g , t ) l ≥ h l \ge h l ≥ h P h ( g , t ) \mathcal{P}_h(g,t) P h ( g , t ) P h ( g , t ) \mathcal{P}_h(g,t) P h ( g , t ) B ( x , r ) B(x,r) B ( x , r ) ∑ l = h H − 1 2 9 κ r / 2 l ≤ 2 10 κ r / 2 h \sum_{l=h}^{H-1}2^9\kappa r/2^l \le2^{10}\kappa r/2^h ∑ l = h H − 1 2 9 κ r / 2 l ≤ 2 10 κ r / 2 h
Let x ≠ y x \ne y x = y r = dist ( x , y ) ≥ 2 r=\operatorname{dist}(x,y)\ge2 r = dist ( x , y ) ≥ 2 Λ r ≥ 2 H \Lambda r\ge2^H Λ r ≥ 2 H ( x , y ) (x,y) ( x , y ) Λ \Lambda Λ h h h 2 h ≤ Λ r < 2 h + 1 2^h\le\Lambda r<2^{h+1} 2 h ≤ Λ r < 2 h + 1 1 ≤ h < H 1\le h<H 1 ≤ h < H ( x , y ) (x,y) ( x , y ) Λ \Lambda Λ lev ( x , y ) > h \operatorname{lev}(x,y)>h lev ( x , y ) > h P h ( g , t ) \mathcal{P}_h(g,t) P h ( g , t ) x , y ∈ B ( x , r ) x,y\in B(x,r) x , y ∈ B ( x , r ) 2 10 κ r / 2 h < 2 11 κ / Λ 2^{10}\kappa r/2^h<2^{11}\kappa/\Lambda 2 10 κ r / 2 h < 2 11 κ /Λ ∣ X ∣ |X| ∣ X ∣ H H H
The Network and the Dynamic Program
This section defines the network in which the algorithm routes the blocks, and shows how to optimize over it. § 7.1 defines the network and proves that it never underestimates distances and that it is accurate for pairs that are not Λ \Lambda Λ § 7.2 defines network solutions, states that an optimal one can be computed by dynamic programming, and converts network solutions into clusterings. § 7.3 gives the dynamic program.
Throughout this section, X = P ∪ F X=P\cup F X = P ∪ F κ \kappa κ 2 2 2 P P P u ≥ 1 u\ge1 u ≥ 1 v ≥ 0 v\ge0 v ≥ 0 D \mathcal{D} D X X X 0 , … , H 0,\ldots,H 0 , … , H λ ∈ ( 0 , 1 ] \lambda\in(0,1] λ ∈ ( 0 , 1 ]
The network
Every cell C C C h h h N C ⊆ C N_C\subseteq C N C ⊆ C portals : a maximal subset of C C C λ 2 h \lambda2^h λ 2 h C C C 2 h 2^h 2 h B ( x , 2 h ) B(x,2^h) B ( x , 2 h ) x ∈ C x\in C x ∈ C r = 2 h r=2^h r = 2 h s = λ 2 h s=\lambda2^h s = λ 2 h
∣ N C ∣ ≤ M : = ⌈ ( 4 λ ) κ ⌉ . (8) |N_C| \le M := \left\lceil\left(\frac{4}{\lambda}\right)^\kappa\right\rceil. \tag*{(8)} ∣ N C ∣ ≤ M := ⌈ ( λ 4 ) κ ⌉ . (8) A cell of level 0 0 0 C C C h < H h<H h < H D D D x ′ ∈ N C x'\in N_C x ′ ∈ N C par ( x ′ ) ∈ N D \operatorname{par}(x')\in N_D par ( x ′ ) ∈ N D dist ( x ′ , par ( x ′ ) ) ≤ λ 2 h + 1 \operatorname{dist}(x',\operatorname{par}(x'))\le\lambda2^{h+1} dist ( x ′ , par ( x ′ )) ≤ λ 2 h + 1 x ′ ∈ D x'\in D x ′ ∈ D N D N_D N D
Definition 7.1 (Network). The network N D \mathcal{N}_{\mathcal{D}} N D x C ↑ x_C^\uparrow x C ↑ x C ↓ x_C^\downarrow x C ↓ C C C x ′ ∈ N C x'\in N_C x ′ ∈ N C x s r c x^{\mathrm{src}} x src x s n k x^{\mathrm{snk}} x snk x ∈ X x\in X x ∈ X
ascending arcs x C ↑ → par ( x ′ ) D ↑ x_C^\uparrow\to\operatorname{par}(x')_D^\uparrow x C ↑ → par ( x ′ ) D ↑ descending arcs par ( x ′ ) D ↓ → x C ↓ \operatorname{par}(x')_D^\downarrow\to x_C^\downarrow par ( x ′ ) D ↓ → x C ↓ 0 0 0 C C C D D D x ′ ∈ N C x'\in N_C x ′ ∈ N C
crossing arcs x C ↑ → y C ↓ x_C^\uparrow\to y_C^\downarrow x C ↑ → y C ↓ ( dist ( x ′ , y ′ ) + 4 λ 2 h ) z (\operatorname{dist}(x',y')+4\lambda2^h)^z ( dist ( x ′ , y ′ ) + 4 λ 2 h ) z C C C h h h x ′ , y ′ ∈ N C x',y'\in N_C x ′ , y ′ ∈ N C
arcs x s r c → x { x } ↑ x^{\mathrm{src}}\to x_{\{x\}}^\uparrow x src → x { x } ↑ x { x } ↓ → x s n k x_{\{x\}}^\downarrow\to x^{\mathrm{snk}} x { x } ↓ → x snk x s r c → x s n k x^{\mathrm{src}}\to x^{\mathrm{snk}} x src → x snk 0 0 0 x ∈ X x\in X x ∈ X { x } \{x\} { x } 0 0 0 x x x
Ascending arcs increase the level, descending arcs decrease it, and crossing arcs go from ascending to descending copies. So N D \mathcal{N}_{D} N D x s r c x^{\mathrm{src}} x src y s n k y^{\mathrm{snk}} y snk x s r c → y s n k x^{\mathrm{src}} \to y^{\mathrm{snk}} x src → y snk x = y x=y x = y
Lemma 7.2 (Routes). Let x , y ∈ X x,y \in X x , y ∈ X
(a) Every path from x s r c x^{\mathrm{src}} x src y s n k y^{\mathrm{snk}} y snk dist ( x , y ) z \operatorname{dist}(x,y)^z dist ( x , y ) z
(b) If x = y x=y x = y ( x , y ) (x,y) ( x , y ) Λ \Lambda Λ x s r c x^{\mathrm{src}} x src y s n k y^{\mathrm{snk}} y snk ( ( 1 + 8 λ Λ ) dist ( x , y ) ) z \bigl((1+8\lambda\Lambda)\operatorname{dist}(x,y)\bigr)^z ( ( 1 + 8 λ Λ ) dist ( x , y ) ) z
Proof. For l = 0 , … , H l=0,\ldots,H l = 0 , … , H C l C_l C l l l l x x x x 0 = x x_0=x x 0 = x x l = par ( x l − 1 ) ∈ N C l x_l=\operatorname{par}(x_{l-1})\in N_{C_l} x l = par ( x l − 1 ) ∈ N C l x { x } s r c x^{\mathrm{src}}_{\{x\}} x { x } src ( x 0 ) C 0 ↑ , ( x 1 ) C 1 ↑ , … (x_0)^{\uparrow}_{C_0},(x_1)^{\uparrow}_{C_1},\ldots ( x 0 ) C 0 ↑ , ( x 1 ) C 1 ↑ , …
dist ( x , x l ) ≤ ∑ t = 1 l λ 2 t < λ 2 l + 1 . \operatorname{dist}(x,x_l)\le\sum_{t=1}^{l}\lambda2^t<\lambda2^{l+1}. dist ( x , x l ) ≤ t = 1 ∑ l λ 2 t < λ 2 l + 1 . Similarly, let C l ′ C'_l C l ′ y y y y 0 = y y_0=y y 0 = y y l = par ( y l − 1 ) y_l=\operatorname{par}(y_{l-1}) y l = par ( y l − 1 ) dist ( y , y l ) < λ 2 l + 1 \operatorname{dist}(y,y_l)<\lambda2^{l+1} dist ( y , y l ) < λ 2 l + 1 y { y } s n k y^{\mathrm{snk}}_{\{y\}} y { y } snk ( y l ) C l ′ ↓ , … , ( y 0 ) C 0 ′ ↓ (y_l)^{\downarrow}_{C'_l},\ldots,(y_0)^{\downarrow}_{C'_0} ( y l ) C l ′ ↓ , … , ( y 0 ) C 0 ′ ↓ l l l
(a) The arc x s r c → y s n k x^{\mathrm{src}}\to y^{\mathrm{snk}} x src → y snk x = y x=y x = y ( x l ) C l ↑ (x_l)^{\uparrow}_{C_l} ( x l ) C l ↑ ( y l ) C l ′ ↓ (y_l)^{\downarrow}_{C'_l} ( y l ) C l ′ ↓ C l ′ = C l C'_l=C_l C l ′ = C l ( dist ( x l , y l ) + 4 λ 2 l ) z (\operatorname{dist}(x_l,y_l)+4\lambda2^l)^z ( dist ( x l , y l ) + 4 λ 2 l ) z dist ( x , y ) ≤ dist ( x , x l ) + dist ( x l , y l ) + dist ( y l , y ) < dist ( x l , y l ) + 4 λ 2 l \operatorname{dist}(x,y)\le\operatorname{dist}(x,x_l)+\operatorname{dist}(x_l,y_l)+\operatorname{dist}(y_l,y)<\operatorname{dist}(x_l,y_l)+4\lambda2^l dist ( x , y ) ≤ dist ( x , x l ) + dist ( x l , y l ) + dist ( y l , y ) < dist ( x l , y l ) + 4 λ 2 l
(b) For x = y x=y x = y x s r c → x s n k x^{\mathrm{src}}\to x^{\mathrm{snk}} x src → x snk l = lev ( x , y ) l=\operatorname{lev}(x,y) l = lev ( x , y ) C l = C l ′ C_l=C'_l C l = C l ′ 2 l ≤ Λ dist ( x , y ) 2^l\le\Lambda\operatorname{dist}(x,y) 2 l ≤ Λ dist ( x , y ) ( x l ) C l ↑ (x_l)^{\uparrow}_{C_l} ( x l ) C l ↑ ( y l ) C l ↓ (y_l)^{\downarrow}_{C_l} ( y l ) C l ↓ y s n k y^{\mathrm{snk}} y snk ( dist ( x l , y l ) + 4 λ 2 l ) z ≤ ( dist ( x , y ) + 8 λ 2 l ) z ≤ ( ( 1 + 8 λ Λ ) dist ( x , y ) ) z (\operatorname{dist}(x_l,y_l)+4\lambda2^l)^z\le(\operatorname{dist}(x,y)+8\lambda2^l)^z\le((1+8\lambda\Lambda)\operatorname{dist}(x,y))^z ( dist ( x l , y l ) + 4 λ 2 l ) z ≤ ( dist ( x , y ) + 8 λ 2 l ) z ≤ (( 1 + 8 λ Λ ) dist ( x , y ) ) z
Network solutions
In a network solution, every block is either rescued or normal. Normal blocks inject their clients as mass at their representatives, and a location that receives t t t ⌈ t / U ⌉ \lceil t/U\rceil ⌈ t / U ⌉
Definition 7.3 (Network solution). A network solution ( R , f ) (R,f) ( R , f ) R ⊆ I R\subseteq I R ⊆ I f ≥ 0 f\ge0 f ≥ 0 N D \mathcal{N}_{D} N D x s r c x^{\mathrm{src}} x src s x : = ∑ i ∉ R : a i = x w i s_x:=\sum_{i\notin R:a_i=x}w_i s x := ∑ i ∈ / R : a i = x w i y s n k y^{\mathrm{snk}} y snk t y t_y t y t y = 0 t_y=0 t y = 0 y ∉ F y\notin F y ∈ / F ∣ R ∣ + ∑ y ∈ F ⌈ t y / U ⌉ |R|+\sum_{y\in F}\lceil t_y/U\rceil ∣ R ∣ + ∑ y ∈ F ⌈ t y / U ⌉ feasible if this number is at most K K K
Φ ( R , f ) : = ∑ i ∉ R ( u − 1 ) A i + v ∑ e cost ( e ) f ( e ) , \Phi(R,f):=\sum_{i\notin R}(u-1)A_i+v\sum_e\operatorname{cost}(e)f(e), Φ ( R , f ) := i ∈ / R ∑ ( u − 1 ) A i + v e ∑ cost ( e ) f ( e ) , where the second sum is over all arcs e e e N D \mathcal{N}_{D} N D
Lemma 7.4 (Dynamic program). A feasible network solution of minimum cost, if one exists, can be computed in time ( 2 n + 1 ) O ( M ) ⋅ poly ( ∣ X ∣ , H ) (2n+1)^{O(M)}\cdot\operatorname{poly}(|X|,H) ( 2 n + 1 ) O ( M ) ⋅ poly ( ∣ X ∣ , H )
Lemma 7.5 (From network solutions to clusterings). Suppose that every block is ( u , v ) (u,v) ( u , v ) ( R , f ) (R,f) ( R , f ) K K K F F F A + Φ ( R , f ) A+\Phi(R,f) A + Φ ( R , f )
Proof. Since N D \mathcal{N}_{D} N D f f f ∑ x s x \sum_{x} s_{x} ∑ x s x x s r c x^{\mathrm{src}} x src y s n k y^{\mathrm{snk}} y snk y ∈ F y \in F y ∈ F y ∈ F y \in F y ∈ F t y > 0 t_{y} > 0 t y > 0 ⌈ t y / U ⌉ \lceil t_{y}/U\rceil ⌈ t y / U ⌉ y y y t y t_{y} t y y s n k y^{\mathrm{snk}} y snk U U U x x x s x s_{x} s x x s r c x^{\mathrm{src}} x src x x x i i i w i w_{i} w i y i , 1 , … , y i , w i y_{i,1}, \ldots, y_{i,w_{i}} y i , 1 , … , y i , w i ∑ r dist ( a i , y i , r ) z \sum_{r}\operatorname{dist}(a_{i},y_{i,r})^{z} ∑ r dist ( a i , y i , r ) z i i i
∑ i ∉ R ∑ r = 1 w i dist ( a i , y i , r ) z ≤ ∑ e cost ( e ) f ( e ) . \sum_{i\notin R}\sum_{r=1}^{w_i}\operatorname{dist}(a_i,y_{i,r})^z \le\sum_e \operatorname{cost}(e)f(e). i ∈ / R ∑ r = 1 ∑ w i dist ( a i , y i , r ) z ≤ e ∑ cost ( e ) f ( e ) . Assign the clients of every normal block i i i u A i + v ∑ r dist ( a i , y i , r ) z uA_i+v\sum_r\operatorname{dist}(a_i,y_{i,r})^z u A i + v ∑ r dist ( a i , y i , r ) z i i i a i a_i a i A i A_i A i U U U K K K ∑ i ∈ R A i + ∑ i ∉ R ( u A i + v ∑ r dist ( a i , y i , r ) z ) ≤ A + Φ ( R , f ) \sum_{i\in R} A_i+\sum_{i\notin R}(uA_i+v\sum_r\operatorname{dist}(a_i,y_{i,r})^z)\le A+\Phi(R,f) ∑ i ∈ R A i + ∑ i ∈ / R ( u A i + v ∑ r dist ( a i , y i , r ) z ) ≤ A + Φ ( R , f )
The dynamic program
We prove Theorem 7.4. The network follows the tree of cells, which gives a tree decomposition of small width.
Tree decompositions. A tree decomposition of a directed graph is a tree together with a set of vertices of the graph, called a bag, for every node of the tree, such that both endpoints of every arc lie in a common bag and, for every vertex, the nodes whose bags contain it form a nonempty subtree. Its width is the largest bag size minus one. A nice tree decomposition with introduce-arc nodes is rooted, has empty bags at the root and at the leaves, and every other node is of one of the following types: an introduce node or a forget node, whose bag differs from the bag of its only child by one vertex; an introduce-arc node, which has the bag of its only child and is labelled with an arc between two bag vertices, every arc being the label of exactly one node; or a join node, which has two children with the same bag as itself. A tree decomposition can be turned in polynomial time into a nice one of the same width whose number of nodes is polynomial in the number of nodes, the width, and the number of arcs [CFK+15, Section 7.3].
A tree decomposition of the network. Start with the tree whose nodes are the cells of D \mathcal{D} D C C C bag ( C ) = { x C ↑ , x C ↓ : x ′ ∈ N C } \operatorname{bag}(C)=\{x_{C}^{\uparrow},x_{C}^{\downarrow}:x'\in N_C\} bag ( C ) = { x C ↑ , x C ↓ : x ′ ∈ N C } C C C D D D bag ( C ) ∪ bag ( D ) \operatorname{bag}(C)\cup\operatorname{bag}(D) bag ( C ) ∪ bag ( D ) { x } \{x\} { x } bag ( { x } ) ∪ { x s r c , x s n k } \operatorname{bag}(\{x\})\cup\{x^{\mathrm{src}},x^{\mathrm{snk}}\} bag ({ x }) ∪ { x src , x snk } bag ( C ) \operatorname{bag}(C) bag ( C ) C C C D D D bag ( C ) ∪ bag ( D ) \operatorname{bag}(C)\cup\operatorname{bag}(D) bag ( C ) ∪ bag ( D ) x x x x C ↑ x_{C}^{\uparrow} x C ↑ x C ↓ x_{C}^{\downarrow} x C ↓ C C C C C C C C C x s r c x^{\mathrm{src}} x src x s n k x^{\mathrm{snk}} x snk N D \mathcal{N}_{D} N D O ( H ∣ X ∣ ) O(H|X|) O ( H ∣ X ∣ ) 4 M 4M 4 M
Local tables. For a location x x x I x = { i : a i = x } I_x=\{i:a_i=x\} I x = { i : a i = x } 0 ≤ s ≤ n 0\le s\le n 0 ≤ s ≤ n 0 ≤ r ≤ K 0\le r\le K 0 ≤ r ≤ K loc x ( s , r ) \operatorname{loc}_{x}(s,r) loc x ( s , r ) ∑ i ∈ I x R x ( u − 1 ) A i \sum_{i\in I_x} R_x(u-1)A_i ∑ i ∈ I x R x ( u − 1 ) A i R x ⊆ I x R_x\subseteq I_x R x ⊆ I x ∣ R x ∣ = r |R_x|=r ∣ R x ∣ = r ∑ i ∈ I x ∖ R x w i = s \sum_{i\in I_x\setminus R_x}w_i=s ∑ i ∈ I x ∖ R x w i = s ∞ \infty ∞ R x R_x R x I x I_x I x loc x ( s , r ) \operatorname{loc}_{x}(s,r) loc x ( s , r ) O ( ∣ I x ∣ n K ) O(|I_x|nK) O ( ∣ I x ∣ n K ) R x R_x R x
States. Consider a node of the nice tree decomposition; the arcs introduced below it and the vertices forgotten below it are fixed. A partial solution at this node assigns a flow value in { 0 , … , n } \{0,\ldots,n\} { 0 , … , n } r x r_x r x x s r c x^{\mathrm{src}} x src x s r c x^{\mathrm{src}} x src s s s loc x ( s , r x ) < ∞ \operatorname{loc}_x(s,r_x)<\infty loc x ( s , r x ) < ∞ y s n k y^{\mathrm{snk}} y snk t y ≥ 0 t_y\ge0 t y ≥ 0 t y = 0 t_y=0 t y = 0 y ∉ F y\notin F y ∈ / F ( b a l , k ′ ) (\mathrm{bal},k') ( bal , k ′ ) b a l \mathrm{bal} bal k ′ = ∑ r x + ∑ ⌈ t y / U ⌉ k'=\sum r_x+\sum\lceil t_y/U\rceil k ′ = ∑ r x + ∑ ⌈ t y / U ⌉ ∑ loc x ( s , r x ) + v ∑ cost ( e ) f ( e ) \sum\operatorname{loc}_x(s,r_x)+v\sum\operatorname{cost}(e)f(e) ∑ loc x ( s , r x ) + v ∑ cost ( e ) f ( e ) { − n , … , n } \{-n,\ldots,n\} { − n , … , n } k ′ ≤ K k'\le K k ′ ≤ K
Transitions. The tables are computed bottom-up. A leaf has the single state with empty b a l \mathrm{bal} bal k ′ = 0 k'=0 k ′ = 0 e e e f ( e ) ∈ { 0 , … , n } f(e)\in\{0,\ldots,n\} f ( e ) ∈ { 0 , … , n } e e e f ( e ) f(e) f ( e ) v ⋅ cost ( e ) ⋅ f ( e ) v\cdot\operatorname{cost}(e)\cdot f(e) v ⋅ cost ( e ) ⋅ f ( e ) x s r c x^{\mathrm{src}} x src x s r c x^{\mathrm{src}} x src − s -s − s s ≥ 0 s\ge0 s ≥ 0 r r r r r r k ′ k' k ′ loc x ( s , r ) \operatorname{loc}_x(s,r) loc x ( s , r ) y s n k y^{\mathrm{snk}} y snk y s n k y^{\mathrm{snk}} y snk t ≥ 0 t\ge0 t ≥ 0 t = 0 t=0 t = 0 y ∉ F y\notin F y ∈ / F ⌈ t / U ⌉ \lceil t/U\rceil ⌈ t / U ⌉ k ′ k' k ′ k ′ k' k ′ k ′ ≤ K k'\le K k ′ ≤ K
Correctness of the value range. In a network solution, f f f n n n n n n n n n { − n , … , n } \{-n,\ldots,n\} { − n , … , n }
Running time. A table has at most ( 2 n + 1 ) 4 M ( K + 1 ) (2n+1)^{4M}(K+1) ( 2 n + 1 ) 4 M ( K + 1 ) O ( n ) O(n) O ( n ) K ≤ 2 n K\le2n K ≤ 2 n poly ( ∣ X ∣ , H ) \operatorname{poly}(|X|,H) poly ( ∣ X ∣ , H ) ( 2 n + 1 ) O ( M ) ⋅ poly ( ∣ X ∣ , H ) (2n+1)^{O(M)}\cdot\operatorname{poly}(|X|,H) ( 2 n + 1 ) O ( M ) ⋅ poly ( ∣ X ∣ , H )
Proof of the Framework Theorem
We prove Theorem 2.6 for steps 1, 3 and 4 of Algorithm 1, applied to the given block partition and the given pair ( u , v ) (u,v) ( u , v )
γ = min { ε , η } 4 , L = ⌈ 12 η ⌉ , k 0 = ⌈ 4 ( L + 1 ) η ⌉ , Λ = 2 13 κ γ , λ = η 8 z Λ . (9) \gamma=\frac{\min\{\varepsilon,\eta\}}{4},\qquad L=\left\lceil\frac{12}{\eta}\right\rceil,\qquad k_0=\left\lceil\frac{4(L+1)}{\eta}\right\rceil,\qquad\Lambda=\frac{2^{13}\kappa}{\gamma},\qquad\lambda=\frac{\eta}{8z\Lambda}. \tag*{(9)} γ = 4 min { ε , η } , L = ⌈ η 12 ⌉ , k 0 = ⌈ η 4 ( L + 1 ) ⌉ , Λ = γ 2 13 κ , λ = 8 z Λ η . (9) Here ∣ γ k ∣ \lvert\gamma k\rvert ∣ γ k ∣ L L L b = ⌈ κ ⌉ − ⌈ γ k ⌉ b=\lceil\kappa\rceil-\lceil\gamma k\rceil b = ⌈ κ ⌉ − ⌈ γ k ⌉ k < k 0 k<k_0 k < k 0 Λ \Lambda Λ 1 + 8 λ Λ = 1 + η / z 1+8\lambda\Lambda=1+\eta/z 1 + 8 λ Λ = 1 + η / z X = P ∪ F X=P\cup F X = P ∪ F H \mathcal{H} H λ \lambda λ
Proof of Theorem 2.6 . If k < k 0 k < k_0 k < k 0 k k k cost ( S ∗ ) \operatorname{cost}(S^*) cost ( S ∗ ) β − ( β − 1 ) ε ≥ 1 \beta- (\beta- 1)\varepsilon\ge1 β − ( β − 1 ) ε ≥ 1 k ≥ k 0 k \ge k_0 k ≥ k 0 S ∗ S^* S ∗ k k k A ≤ ( 1 + η ) cost ( S ∗ ) A \le(1 + \eta)\operatorname{cost}(S^*) A ≤ ( 1 + η ) cost ( S ∗ ) π \pi π S ∗ S^* S ∗ T i T_i T i W i = ( u − 1 ) A i + v T i W_i = (u - 1)A_i + vT_i W i = ( u − 1 ) A i + v T i ∑ i W i ≤ ( β − 1 ) ( 1 + η ) cost ( S ∗ ) \sum_i W_i \le(\beta- 1)(1 + \eta)\operatorname{cost}(S^*) ∑ i W i ≤ ( β − 1 ) ( 1 + η ) cost ( S ∗ ) (3) . The proof has three steps: choose the rescued blocks, find a decomposition with few bad edges, and route the remaining mass through the network.
Rescued blocks. Apply Theorem 2.5 to π \pi π L L L (9) and b = ⌊ ε k ⌋ − ⌊ γ k ⌋ b = \lfloor\varepsilon k\rfloor- \lfloor\gamma k\rfloor b = ⌊ ε k ⌋ − ⌊ γ k ⌋ 0 ≤ b ≤ k 0 \le b \le k 0 ≤ b ≤ k γ ≤ ε ≤ 1 \gamma\le\varepsilon\le1 γ ≤ ε ≤ 1 R 0 ⊆ I R_0 \subseteq I R 0 ⊆ I J 0 ⊆ J J_0 \subseteq J J 0 ⊆ J b ≥ ε k − γ k − 1 b \ge\varepsilon k - \gamma k - 1 b ≥ ε k − γ k − 1 γ ≤ η / 4 \gamma\le\eta/4 γ ≤ η /4 3 / ( L + 1 ) ≤ η / 4 3/(L + 1) \le\eta/4 3/ ( L + 1 ) ≤ η /4 ( L + 1 ) / k ≤ η / 4 (L + 1)/k \le\eta/4 ( L + 1 ) / k ≤ η /4
b − L k − 3 L + 1 ≥ ε − γ − L + 1 k − 3 L + 1 ≥ ε − η , \frac{b-L}{k}-\frac{3}{L+1}\ge\varepsilon-\gamma-\frac{L+1}{k}-\frac{3}{L+1}\ge\varepsilon-\eta, k b − L − L + 1 3 ≥ ε − γ − k L + 1 − L + 1 3 ≥ ε − η , so ∑ i ∈ R 0 W i ≥ ( ε − η ) ∑ i W i \sum_{i\in R_0} W_i \ge(\varepsilon-\eta)\sum_i W_i ∑ i ∈ R 0 W i ≥ ( ε − η ) ∑ i W i
Few bad edges. Let E E E ( a i , c j ) (a_i,c_j) ( a i , c j ) i j ij ij supp ( π ) \operatorname{supp}(\pi) supp ( π ) ∣ I ∣ + ∣ J ∣ < 3 k |I| + |J| < 3k ∣ I ∣ + ∣ J ∣ < 3 k ∣ E ∣ < 3 k |E| < 3k ∣ E ∣ < 3 k Λ \Lambda Λ D ∈ H D \in\mathcal{H} D ∈ H 2 11 κ ∣ E ∣ / Λ < 3 γ k / 4 2^{11}\kappa|E|/\Lambda< 3\gamma k/4 2 11 κ ∣ E ∣/Λ < 3 γ k /4 E E E Λ \Lambda Λ γ k \gamma k γ k ⌊ γ k ⌋ \lfloor\gamma k\rfloor ⌊ γ k ⌋ R b a d R_{\mathrm{bad}} R bad ∣ R b a d ∣ ≤ ⌊ γ k ⌋ |R_{\mathrm{bad}}| \le\lfloor\gamma k\rfloor ∣ R bad ∣ ≤ ⌊ γ k ⌋ R = R 0 ∪ R b a d R = R_0 \cup R_{\mathrm{bad}} R = R 0 ∪ R bad
Routing. We define a network solution ( R , f ) (R,f) ( R , f ) N D N_{\mathcal{D}} N D i j ij ij supp ( π ) \operatorname{supp}(\pi) supp ( π ) i ∉ R i \notin R i ∈ / R ( a i , c j ) (a_i,c_j) ( a i , c j ) Λ \Lambda Λ π i j \pi_{ij} π ij ( a i ) s r c (a_i)^{\mathrm{src}} ( a i ) src ( c j ) s n k (c_j)^{\mathrm{snk}} ( c j ) snk x s r c x^{\mathrm{src}} x src ∑ i ∉ R : a i = x w i = s x \sum_{i\notin R:a_i=x}w_i=s_x ∑ i ∈ / R : a i = x w i = s x ℓ j ′ = ∑ i ∉ R π i j ≤ ℓ j ≤ U \ell'_j=\sum_{i\notin R}\pi_{ij}\le\ell_j\le U ℓ j ′ = ∑ i ∈ / R π ij ≤ ℓ j ≤ U j ∈ J j\in J j ∈ J ℓ j ′ = 0 \ell'_j=0 ℓ j ′ = 0 j ∈ J 0 j\in J_0 j ∈ J 0 t y t_y t y y s n k y^{\mathrm{snk}} y snk ℓ j ′ \ell'_j ℓ j ′ j j j c j = y c_j=y c j = y ⌈ t y / U ⌉ \lceil t_y/U\rceil ⌈ t y / U ⌉ ℓ j ′ > 0 \ell'_j>0 ℓ j ′ > 0
∣ R ∣ + ∑ y ∈ F ⌈ t y U ⌉ ≤ ∣ R b a d ∣ + ∣ R 0 ∣ + ( ∣ J ∣ − ∣ J 0 ∣ ) ≤ ⌊ γ k ⌋ + k + b = K , |R|+\sum_{y\in F}\left\lceil\frac{t_y}{U}\right\rceil\le|R_{\mathrm{bad}}|+|R_0|+(|J|-|J_0|)\le\lfloor\gamma k\rfloor+k+b=K, ∣ R ∣ + y ∈ F ∑ ⌈ U t y ⌉ ≤ ∣ R bad ∣ + ∣ R 0 ∣ + ( ∣ J ∣ − ∣ J 0 ∣ ) ≤ ⌊ γ k ⌋ + k + b = K , so ( R , f ) (R,f) ( R , f ) i j ij ij ( ( 1 + η / z ) dist ( a i , c j ) ) z ≤ e η dist ( a i , c j ) z ((1+\eta/z)\operatorname{dist}(a_i,c_j))^z\le e^\eta\operatorname{dist}(a_i,c_j)^z (( 1 + η / z ) dist ( a i , c j ) ) z ≤ e η dist ( a i , c j ) z
Φ ( R , f ) ≤ ∑ i ∉ R ( ( u − 1 ) A i + e η v T i ) ≤ e η ∑ i ∉ R 0 W i ≤ e η ( 1 − ε + η ) ∑ i W i . \Phi(R,f)\le\sum_{i\notin R}\bigl((u-1)A_i+e^\eta vT_i\bigr)\le e^\eta\sum_{i\notin R_0}W_i\le e^\eta(1-\varepsilon+\eta)\sum_i W_i. Φ ( R , f ) ≤ i ∈ / R ∑ ( ( u − 1 ) A i + e η v T i ) ≤ e η i ∈ / R 0 ∑ W i ≤ e η ( 1 − ε + η ) i ∑ W i . Conclusion. The algorithm finds a network solution of cost at most Φ ( R , f ) \Phi(R,f) Φ ( R , f ) (3) its output costs at most
A + e η ( 1 + η ) ( 1 − ε + η ) ( β − 1 ) cost ( S ∗ ) . A+e^\eta(1+\eta)(1-\varepsilon+\eta)(\beta-1)\operatorname{cost}(S^*). A + e η ( 1 + η ) ( 1 − ε + η ) ( β − 1 ) cost ( S ∗ ) . For η ≤ 1 / 10 \eta\le1/10 η ≤ 1/10 e η ( 1 + η ) ≤ 1 + 2.2 η e^\eta(1+\eta)\le1+2.2\eta e η ( 1 + η ) ≤ 1 + 2.2 η e η ( 1 + η ) ( 1 − ε + η ) ≤ 1 − ε + 3.5 η e^\eta(1+\eta)(1-\varepsilon+\eta)\le1-\varepsilon+3.5\eta e η ( 1 + η ) ( 1 − ε + η ) ≤ 1 − ε + 3.5 η A ≤ ( 1 + η ) cost ( S ∗ ) A\le(1+\eta)\operatorname{cost}(S^*) A ≤ ( 1 + η ) cost ( S ∗ )
( 1 + η + ( β − 1 ) ( 1 − ε ) + 3.5 ( β − 1 ) η ) cost ( S ∗ ) ≤ ( β − ( β − 1 ) ε + 4 β η ) cost ( S ∗ ) . \bigl(1+\eta+(\beta-1)(1-\varepsilon)+3.5(\beta-1)\eta\bigr)\operatorname{cost}(S^*)\le\bigl(\beta-(\beta-1)\varepsilon+4\beta\eta\bigr)\operatorname{cost}(S^*). ( 1 + η + ( β − 1 ) ( 1 − ε ) + 3.5 ( β − 1 ) η ) cost ( S ∗ ) ≤ ( β − ( β − 1 ) ε + 4 β η ) cost ( S ∗ ) . Running time. For k < k 0 k<k_0 k < k 0 ( ∣ F ∣ + 1 ) k 0 (|F|+1)^{k_0} ( ∣ F ∣ + 1 ) k 0 H \mathcal{H} H O ( ∣ X ∣ 7 ) O(|X|^7) O ( ∣ X ∣ 7 ) ( 2 n + 1 ) O ( M ) poly ( ∣ X ∣ , H ) (2n+1)^{O(M)}\operatorname{poly}(|X|,H) ( 2 n + 1 ) O ( M ) poly ( ∣ X ∣ , H ) H = O ( log Δ ( X ) ) H=O(\log\Delta(X)) H = O ( log Δ ( X )) M = ⌈ ( 4 / λ ) κ ⌉ M=\lceil(4/\lambda)\kappa\rceil M = ⌈( 4/ λ ) κ ⌉ (9) , 4 / λ = 2 18 z κ / ( η γ ) 4/\lambda=2^{18}z\kappa/(\eta\gamma) 4/ λ = 2 18 z κ / ( η γ )
M = ( z κ η min { ε , η } ) O ( κ ) . M=\left(\frac{z\kappa}{\eta\min\{\varepsilon,\eta\}}\right)^{O(\kappa)}. M = ( η min { ε , η } z κ ) O ( κ ) . Algorithm 1 also runs the local search of Theorem 4.3 , whose swaps involve κ O ( κ ) ( 2 z / η ) O ( 2 z κ / η ) \kappa^{O(\kappa)}(2^z/\eta)^{O(2^z\kappa/\eta)} κ O ( κ ) ( 2 z / η ) O ( 2 z κ / η ) k 0 = O ( 1 / η 2 ) k_0 = O(1/\eta^2) k 0 = O ( 1/ η 2 )
( 2 z κ η min { ε , η } ) O ( 2 z κ / η ) . \left(\frac{2^z\kappa}{\eta\min\{\varepsilon,\eta\}}\right)^{O(2^z\kappa/\eta)}. ( η min { ε , η } 2 z κ ) O ( 2 z κ / η ) . For Theorem 1.2 with P ∪ F P \cup F P ∪ F η \eta η δ / 3 z \delta/3^z δ / 3 z z z z δ ≤ 1 \delta\le1 δ ≤ 1 ( κ / ( ε δ ) ) O ( κ / δ ) (\kappa/(\varepsilon\delta))^{O(\kappa/\delta)} ( κ / ( ε δ ) ) O ( κ / δ ) P P P F F F κ \kappa κ κ ′ \kappa' κ ′ κ \kappa κ § 9.1 and § 10.3 ); for fixed z z z δ ≤ 1 / 2 \delta\le1/2 δ ≤ 1/2 κ ′ = ( 1 / δ ) O ( κ ) \kappa'=(1/\delta)^{O(\kappa)} κ ′ = ( 1/ δ ) O ( κ ) ( κ ′ / ( ε δ ) ) O ( κ ′ / δ ) = ( 1 / ( ε δ ) ) ( 1 / δ ) O ( κ ) (\kappa'/(\varepsilon\delta))^{O(\kappa'/\delta)}=(1/(\varepsilon\delta))^{(1/\delta)^{O(\kappa)}} ( κ ′ / ( ε δ ) ) O ( κ ′ / δ ) = ( 1/ ( ε δ ) ) ( 1/ δ ) O ( κ ) Theorem 1.1 , the candidate centers multiply the exponent of n n n 1 / δ 1/\delta 1/ δ z z z
Proof of Theorem 1.2 for joint low dimension
Suppose that P ∪ F P \cup F P ∪ F κ \kappa κ η = min { 1 / 10 , δ / ( 4 ⋅ 3 z ) } \eta=\min\{1/10,\delta/(4\cdot3^z)\} η = min { 1/10 , δ / ( 4 ⋅ 3 z )} Theorem 4.3 and § 2.1 , compute a block partition with A ≤ ( 1 + η ) O P T A\le(1+\eta)\mathrm{OPT} A ≤ ( 1 + η ) OPT ( 3 z − 1 , ( 3 / 2 ) z − 1 ) (3^{z-1},(3/2)^{z-1}) ( 3 z − 1 , ( 3/2 ) z − 1 ) β = 3 z \beta=3^z β = 3 z Theorem 2.6 with an optimal soft solution as S ∗ S^* S ∗ K K K ( 3 z − ( 3 z − 1 ) ε + 4 ⋅ 3 z η ) O P T ≤ ( 3 z − ( 3 z − 1 ) ε + δ ) O P T (3^z-(3^z-1)\varepsilon+4\cdot3^z\eta)\mathrm{OPT}\le(3^z-(3^z-1)\varepsilon+\delta)\mathrm{OPT} ( 3 z − ( 3 z − 1 ) ε + 4 ⋅ 3 z η ) OPT ≤ ( 3 z − ( 3 z − 1 ) ε + δ ) OPT
One-Sided Low Dimension
The framework theorem needs P ∪ F P \cup F P ∪ F § 10 ), where the other side is a set of candidate centers.
Lemma 9.1 (Compression). Let Q Q Q Y Y Y Q Q Q κ ≥ 1 \kappa\ge1 κ ≥ 1 η ∈ ( 0 , 1 / 4 ] \eta\in(0,1/4] η ∈ ( 0 , 1/4 ] μ : Y → Y \mu:Y\to Y μ : Y → Y y y y μ ( y ) \mu(y) μ ( y ) x ∈ Q x\in Q x ∈ Q y ∈ Y y\in Y y ∈ Y
dist ( x , μ ( y ) ) ≤ ( 1 + 3 η ) dist ( x , y ) and dist ( x , y ) ≤ ( 1 + 3 η ) dist ( x , μ ( y ) ) , (10) \operatorname{dist}(x,\mu(y))\le(1+3\eta)\operatorname{dist}(x,y)
\quad\text{and}\quad
\operatorname{dist}(x,y)\le(1+3\eta)\operatorname{dist}(x,\mu(y)), \tag*{(10)} dist ( x , μ ( y )) ≤ ( 1 + 3 η ) dist ( x , y ) and dist ( x , y ) ≤ ( 1 + 3 η ) dist ( x , μ ( y )) , (10) and Q ∪ μ ( Y ) Q\cup\mu(Y) Q ∪ μ ( Y ) κ η = O ( ( 32 / η ) κ log 2 ( 2 / η ) ) \kappa_\eta=O((32/\eta)^\kappa\log^2(2/\eta)) κ η = O (( 32/ η ) κ log 2 ( 2/ η ))
By (10) with x = y x=y x = y Q ∩ Y Q\cap Y Q ∩ Y Figure 4 illustrates the construction.
Proof. The construction uses that a point y y y r r r Q Q Q 3 r 3r 3 r q q q x ∈ Q x \in Q x ∈ Q B ( q , 8 r ) B(q,8r) B ( q , 8 r ) dist ( x , y ) \operatorname{dist}(x,y) dist ( x , y ) 2 2 2 dist ( x , q ) \operatorname{dist}(x,q) dist ( x , q ) x x x dist ( x , q ) \operatorname{dist}(x,q) dist ( x , q ) 16 r / η 16r/\eta 16 r / η
Construction. Points y ∈ Y y \in Y y ∈ Y dist ( y , Q ) = 0 \operatorname{dist}(y,Q)=0 dist ( y , Q ) = 0 Q Q Q μ ( y ) = y \mu(y)=y μ ( y ) = y r min r_{\min} r m i n dist ( y , Q ) \operatorname{dist}(y,Q) dist ( y , Q ) y y y r = 2 j r min r=2^j r_{\min} r = 2 j r m i n j ≥ 0 j \ge0 j ≥ 0 r ≤ dist ( y , Q ) < 2 r r \le\operatorname{dist}(y,Q)<2r r ≤ dist ( y , Q ) < 2 r r r r Q r Q_r Q r Q Q Q η r \eta r η r y y y r r r x y x_y x y Q Q Q y y y q ( y ) ∈ Q r q(y) \in Q_r q ( y ) ∈ Q r dist ( x y , q ( y ) ) ≤ η r \operatorname{dist}(x_y,q(y)) \le\eta r dist ( x y , q ( y )) ≤ η r
dist ( y , q ( y ) ) ≤ dist ( y , Q ) + η r < 3 r . (11) \operatorname{dist}(y,q(y)) \le\operatorname{dist}(y,Q)+\eta r<3r. \tag*{(11)} dist ( y , q ( y )) ≤ dist ( y , Q ) + η r < 3 r . (11) For a layer r r r q ∈ Q r q \in Q_r q ∈ Q r Q ∩ B ( q , 8 r ) Q \cap B(q,8r) Q ∩ B ( q , 8 r ) j = 1 , … , ⌈ log 2 ( 2 / η ) ⌉ j=1,\ldots,\lceil\log_2(2/\eta)\rceil j = 1 , … , ⌈ log 2 ( 2/ η )⌉ { x ∈ Q : 2 j + 2 r ≤ dist ( x , q ) < 2 j + 3 r } \{x \in Q:2^{j+2}r \le\operatorname{dist}(x,q)<2^{j+3}r\} { x ∈ Q : 2 j + 2 r ≤ dist ( x , q ) < 2 j + 3 r } η r \eta r η r j j j signature of a point y y y r r r q ( y ) = q q(y)=q q ( y ) = q ⌊ dist ( y , x ) / ( η r ) ⌋ \lfloor\operatorname{dist}(y,x)/(\eta r)\rfloor ⌊ dist ( y , x ) / ( η r )⌋ x x x ⌊ dist ( y , x ) / ( 2 j + 1 η r ) ⌋ \lfloor\operatorname{dist}(y,x)/(2^{j+1}\eta r)\rfloor ⌊ dist ( y , x ) / ( 2 j + 1 η r )⌋ x x x j j j j j j group consists of the points of Y Y Y r r r q ( y ) q(y) q ( y ) μ \mu μ
Distances. Let y , y ′ y,y' y , y ′ r r r q = q ( y ) = q ( y ′ ) q=q(y)=q(y') q = q ( y ) = q ( y ′ ) (11) , dist ( y , y ′ ) < 6 r \operatorname{dist}(y,y')<6r dist ( y , y ′ ) < 6 r ∣ dist ( x , y ) − dist ( x , y ′ ) ∣ ≤ 3 η min { dist ( x , y ) , dist ( x , y ′ ) } |\operatorname{dist}(x,y)-\operatorname{dist}(x,y')| \le3\eta\min\{\operatorname{dist}(x,y),\operatorname{dist}(x,y')\} ∣ dist ( x , y ) − dist ( x , y ′ ) ∣ ≤ 3 η min { dist ( x , y ) , dist ( x , y ′ )} x ∈ Q x \in Q x ∈ Q (10) . If dist ( x , q ) ≤ 8 r \operatorname{dist}(x,q) \le8r dist ( x , q ) ≤ 8 r x ′ x' x ′ dist ( x , x ′ ) ≤ η r \operatorname{dist}(x,x') \le\eta r dist ( x , x ′ ) ≤ η r ∣ dist ( x ′ , y ) − dist ( x ′ , y ′ ) ∣ < η r |\operatorname{dist}(x',y)-\operatorname{dist}(x',y')|<\eta r ∣ dist ( x ′ , y ) − dist ( x ′ , y ′ ) ∣ < η r ∣ dist ( x , y ) − dist ( x , y ′ ) ∣ < 3 η r |\operatorname{dist}(x,y)-\operatorname{dist}(x,y')|<3\eta r ∣ dist ( x , y ) − dist ( x , y ′ ) ∣ < 3 η r dist ( x , y ) \operatorname{dist}(x,y) dist ( x , y ) dist ( x , y ′ ) \operatorname{dist}(x,y') dist ( x , y ′ ) r r r Q Q Q x x x j j j ∣ dist ( x , y ) − dist ( x , y ′ ) ∣ < 3 ⋅ 2 j + 1 η r |\operatorname{dist}(x,y)-\operatorname{dist}(x,y')|<3\cdot2^{j+1}\eta r ∣ dist ( x , y ) − dist ( x , y ′ ) ∣ < 3 ⋅ 2 j + 1 η r dist ( x , y ) \operatorname{dist}(x,y) dist ( x , y ) dist ( x , y ′ ) \operatorname{dist}(x,y') dist ( x , y ′ ) 2 j + 2 r − 3 r ≥ 2 j + 1 r 2^{j+2}r-3r \ge2^{j+1}r 2 j + 2 r − 3 r ≥ 2 j + 1 r x x x dist ( x , q ) ≥ 16 r / η \operatorname{dist}(x,q) \ge16r/\eta dist ( x , q ) ≥ 16 r / η dist ( x , y ) \operatorname{dist}(x,y) dist ( x , y ) dist ( x , y ′ ) \operatorname{dist}(x,y') dist ( x , y ′ ) 16 r / η − 3 r ≥ 2 r / η 16r/\eta-3r \ge2r/\eta 16 r / η − 3 r ≥ 2 r / η ∣ dist ( x , y ) − dist ( x , y ′ ) ∣ ≤ dist ( y , y ′ ) < 6 r |\operatorname{dist}(x,y)-\operatorname{dist}(x,y')| \le\operatorname{dist}(y,y')<6r ∣ dist ( x , y ) − dist ( x , y ′ ) ∣ ≤ dist ( y , y ′ ) < 6 r
Doubling dimension. A point q q q μ ( Y ) ∩ Q \mu(Y) \cap Q μ ( Y ) ∩ Q r r r q ( y ) q(y) q ( y ) ( 32 / η ) κ (32/\eta)^\kappa ( 32/ η ) κ ( 16 / η ) κ (16/\eta)^\kappa ( 16/ η ) κ 11 / η 11/\eta 11/ η dist ( y , x ) < 3 r + 8 r \operatorname{dist}(y,x)<3r+8r dist ( y , x ) < 3 r + 8 r x x x dist ( y , x ) < 3 r + 2 j + 3 r ≤ 19 4 ⋅ 2 j + 1 r \operatorname{dist}(y,x)<3r+2^{j+3}r \le\frac{19}{4}\cdot2^{j+1}r dist ( y , x ) < 3 r + 2 j + 3 r ≤ 4 19 ⋅ 2 j + 1 r x x x j j j q ∈ Q r q \in Q_r q ∈ Q r q ( y ) q(y) q ( y )
N η : = ( 1 + 11 / η ) ( 32 / η ) κ + ⌈ log 2 ( 2 / η ) ⌉ ( 16 / η ) κ N_\eta:= (1+11/\eta)^{(32/\eta)^\kappa+\lceil\log_2(2/\eta)\rceil(16/\eta)^\kappa} N η := ( 1 + 11/ η ) ( 32/ η ) κ + ⌈ l o g 2 ( 2/ η )⌉ ( 16/ η ) κ points y ∈ μ ( Y ) y \in\mu(Y) y ∈ μ ( Y ) r r r
Now consider a ball B ( y 0 , R ) B(y_0,R) B ( y 0 , R ) Q ∪ μ ( Y ) Q \cup\mu(Y) Q ∪ μ ( Y ) x 0 x_0 x 0 Q Q Q y 0 y_0 y 0 R / 2 R/2 R /2 Q ∪ μ ( Y ) Q \cup\mu(Y) Q ∪ μ ( Y )
Case dist ( y 0 , Q ) ≤ 2 R \operatorname{dist}(y_0,Q) \le2R dist ( y 0 , Q ) ≤ 2 R x x x dist ( x , Q ) ≤ R / 8 \operatorname{dist}(x,Q) \le R/8 dist ( x , Q ) ≤ R /8 Q Q Q R / 8 R/8 R /8 B ( x 0 , R / 8 + R + 2 R ) B(x_0,R/8+R+2R) B ( x 0 , R /8 + R + 2 R ) Q Q Q 2 4 κ 2^{4\kappa} 2 4 κ R / 4 R/4 R /4 Q Q Q R / 2 R/2 R /2 x x x x x x μ ( Y ) \mu(Y) μ ( Y ) r r r R / 16 < r ≤ dist ( x , Q ) ≤ 3 R R/16<r\le\operatorname{dist}(x,Q)\le3R R /16 < r ≤ dist ( x , Q ) ≤ 3 R r r r q ( x ) q(x) q ( x ) Q r ∩ B ( x 0 , 3 r + R + 2 R ) ⊆ Q r ∩ B ( x 0 , 12 R ) Q_r\cap B(x_0,3r+R+2R)\subseteq Q_r\cap B(x_0,12R) Q r ∩ B ( x 0 , 3 r + R + 2 R ) ⊆ Q r ∩ B ( x 0 , 12 R ) η r > η R / 16 \eta r>\eta R/16 η r > η R /16 ( 768 / η ) κ (768/\eta)^\kappa ( 768/ η ) κ ( 768 / η ) κ N η (768/\eta)^\kappa N_\eta ( 768/ η ) κ N η x x x R / 2 R/2 R /2
Case dist ( y 0 , Q ) > 2 R \operatorname{dist}(y_0,Q)>2R dist ( y 0 , Q ) > 2 R dist ( ⋅ , Q ) \operatorname{dist}(\cdot,Q) dist ( ⋅ , Q ) 1 1 1 x x x dist ( y 0 , Q ) / 2 < dist ( x , Q ) < 3 dist ( y 0 , Q ) / 2 \operatorname{dist}(y_0,Q)/2<\operatorname{dist}(x,Q)<3\operatorname{dist}(y_0,Q)/2 dist ( y 0 , Q ) /2 < dist ( x , Q ) < 3 dist ( y 0 , Q ) /2 x ∉ Q x\notin Q x ∈ / Q dist ( y 0 , Q ) / 4 \operatorname{dist}(y_0,Q)/4 dist ( y 0 , Q ) /4 3 dist ( y 0 , Q ) / 2 3\operatorname{dist}(y_0,Q)/2 3 dist ( y 0 , Q ) /2 q ( x ) q(x) q ( x ) B ( x 0 , 6 dist ( y 0 , Q ) ) B(x_0,6\operatorname{dist}(y_0,Q)) B ( x 0 , 6 dist ( y 0 , Q )) η dist ( y 0 , Q ) / 4 \eta\operatorname{dist}(y_0,Q)/4 η dist ( y 0 , Q ) /4 ( 96 / η ) κ N η (96/\eta)^{\kappa}N_\eta ( 96/ η ) κ N η R / 2 R/2 R /2 2 4 κ + 6 ( 768 / η ) κ N η 2^{4\kappa}+6(768/\eta)^{\kappa}N_\eta 2 4 κ + 6 ( 768/ η ) κ N η R / 2 R/2 R /2 Q ∪ μ ( Y ) Q\cup\mu(Y) Q ∪ μ ( Y ) κ η = log 2 ( 2 4 κ + 6 ( 768 / η ) κ N η ) = O ( ( 32 / η ) κ log 2 ( 2 / η ) ) \kappa_\eta=\log_2(2^{4\kappa}+6(768/\eta)^{\kappa}N_\eta)=O((32/\eta)^\kappa\log^2(2/\eta)) κ η = log 2 ( 2 4 κ + 6 ( 768/ η ) κ N η ) = O (( 32/ η ) κ log 2 ( 2/ η )) log N η = O ( ( 32 / η ) κ log 2 ( 2 / η ) ) \log N_\eta=O((32/\eta)^\kappa\log^2(2/\eta)) log N η = O (( 32/ η ) κ log 2 ( 2/ η ))
Proof of Theorem 1.2 for one-sided low dimension
The case that P ∪ F P\cup F P ∪ F κ \kappa κ § 8.1 . We may assume δ ≤ 1 \delta\leq1 δ ≤ 1 η c = min { 1 / 4 , δ / ( 36 z ⋅ 3 z ) } \eta_c=\min\{1/4,\delta/(36z\cdot3^z)\} η c = min { 1/4 , δ / ( 36 z ⋅ 3 z )} ( 1 + 3 η c ) 2 z ≤ e 6 z η c ≤ 1 + δ / ( 3 ⋅ 3 z ) (1+3\eta_c)^{2z}\leq e^{6z\eta_c}\leq1+\delta/(3\cdot3^z) ( 1 + 3 η c ) 2 z ≤ e 6 z η c ≤ 1 + δ / ( 3 ⋅ 3 z ) κ ′ = κ η c \kappa'=\kappa_{\eta_c} κ ′ = κ η c
Suppose that the clients have doubling dimension at most κ \kappa κ Q Q Q Y = F Y=F Y = F η c \eta_c η c F ′ = μ ( F ) ⊆ F F'=\mu(F)\subseteq F F ′ = μ ( F ) ⊆ F y y y μ ( y ) \mu(y) μ ( y ) F ′ F' F ′ ( 1 + 3 η c ) z OPT (1+3\eta_c)^z\operatorname{OPT} ( 1 + 3 η c ) z OPT P ∪ F ′ P\cup F' P ∪ F ′ κ ′ \kappa' κ ′ § 8.1 with accuracy δ / 3 \delta/3 δ /3 K K K F ′ ⊆ F F'\subseteq F F ′ ⊆ F ( 3 z − ( 3 z − 1 ) ϵ + δ / 3 ) ( 1 + 3 η c ) z OPT ≤ ( 3 z − ( 3 z − 1 ) ϵ + δ ) OPT (3^z-(3^z-1)\epsilon+\delta/3)(1+3\eta_c)^z\operatorname{OPT}\leq(3^z-(3^z-1)\epsilon+\delta)\operatorname{OPT} ( 3 z − ( 3 z − 1 ) ϵ + δ /3 ) ( 1 + 3 η c ) z OPT ≤ ( 3 z − ( 3 z − 1 ) ϵ + δ ) OPT
Suppose that F F F κ \kappa κ Q = F Q=F Q = F Y Y Y p p p μ ( p ) \mu(p) μ ( p ) P ′ ∪ F P'\cup F P ′ ∪ F κ ′ \kappa' κ ′ ( 1 + 3 η c ) z OPT (1+3\eta_c)^z\operatorname{OPT} ( 1 + 3 η c ) z OPT § 8.1 with accuracy δ / 3 \delta/3 δ /3 (10) , this multiplies the distance of every client to its center by at most 1 + 3 η c 1+3\eta_c 1 + 3 η c ( 3 z − ( 3 z − 1 ) ϵ + δ / 3 ) ( 1 + 3 η c ) 2 z OPT ≤ ( 3 z − ( 3 z − 1 ) ϵ + δ ) OPT (3^z-(3^z-1)\epsilon+\delta/3)(1+3\eta_c)^{2z}\operatorname{OPT}\leq(3^z-(3^z-1)\epsilon+\delta)\operatorname{OPT} ( 3 z − ( 3 z − 1 ) ϵ + δ /3 ) ( 1 + 3 η c ) 2 z OPT ≤ ( 3 z − ( 3 z − 1 ) ϵ + δ ) OPT 3 z 3^z 3 z
Euclidean Space
This section proves Theorem 1.1 . § 10.1 shows that certificates make blocks movable when their representatives are optimal or nearly optimal centers, § 10.2 constructs candidate centers, and § 10.3 assembles the algorithm. For a finite multiset Q Q Q R d \mathbb{R}^d R d Ψ Q ( c ) = ∑ p ∈ Q ∥ p − c ∥ z \Psi_Q(c)=\sum_{p\in Q}\lVert p-c\rVert^z Ψ Q ( c ) = ∑ p ∈ Q ∥ p − c ∥ z Ψ Q \Psi_Q Ψ Q optimal center of Q Q Q z > 1 z>1 z > 1 Ψ Q \Psi_Q Ψ Q
∇ Ψ Q ( c ) = − z ∑ p ∈ Q ∥ p − c ∥ z − 2 ( p − c ) , \nabla\Psi_Q(c)=-z\sum_{p\in Q}\lVert p-c\rVert^{z-2}(p-c), ∇ Ψ Q ( c ) = − z p ∈ Q ∑ ∥ p − c ∥ z − 2 ( p − c ) , where terms with p = c p=c p = c
Fact 10.1. Every finite subset X X X R d \mathbb{R}^d R d 3 d 3d 3 d
Proof. Let Y Y Y B ( x , r ) B(x,r) B ( x , r ) r / 2 r/2 r /2 r / 4 r/4 r /4 Y Y Y 5 r / 4 5r/4 5 r /4 ∣ Y ∣ ≤ 5 d ≤ 2 3 d |Y|\leq5^d\leq2^{3d} ∣ Y ∣ ≤ 5 d ≤ 2 3 d B ( y , r / 2 ) B(y,r/2) B ( y , r /2 ) y ∈ Y y\in Y y ∈ Y B ( x , r ) B(x,r) B ( x , r )
Certificates and movability
Recall from (5) that ( u , g , v ) ∈ R 3 (u,g,v)\in\mathbb{R}^3 ( u , g , v ) ∈ R 3 z z z ∣ 1 + t ∣ z ≤ u + g t + v ∣ t ∣ z |1+t|^z\leq u+gt+v|t|^z ∣1 + t ∣ z ≤ u + g t + v ∣ t ∣ z t ∈ R t\in\mathbb{R} t ∈ R z = 1 z=1 z = 1 ( 1 , 1 ) (1,1) ( 1 , 1 ) z > 1 z>1 z > 1
Lemma 10.2 (Certificates extend to vectors). Let z > 1 z>1 z > 1 ( u , g , v ) (u,g,v) ( u , g , v ) z z z u ≥ 1 u\ge1 u ≥ 1 v ≥ 1 v\ge1 v ≥ 1 p , a , y ∈ R d p,a,y\in\mathbb{R}^{d} p , a , y ∈ R d
∥ p − y ∥ z ≤ u ∥ p − a ∥ z + g ∥ p − a ∥ z − 2 ⟨ p − a , a − y ⟩ + v ∥ a − y ∥ z . \lVert p-y\rVert^{z}\le u\lVert p-a\rVert^{z}+g\lVert p-a\rVert^{z-2}\langle p-a,a-y\rangle+v\lVert a-y\rVert^{z}. ∥ p − y ∥ z ≤ u ∥ p − a ∥ z + g ∥ p − a ∥ z − 2 ⟨ p − a , a − y ⟩ + v ∥ a − y ∥ z . where the middle term is zero if p = a p=a p = a
Proof. The certificate inequality at t = 0 t=0 t = 0 u ≥ 1 u\ge1 u ≥ 1 ∣ t ∣ z \lvert t\rvert^{z} ∣ t ∣ z t → ∞ t\to\infty t → ∞ v ≥ 1 v\ge1 v ≥ 1 p = a p=a p = a v ≥ 1 v\ge1 v ≥ 1 z z z ∥ p − a ∥ = 1 \lVert p-a\rVert=1 ∥ p − a ∥ = 1 t = ∥ a − y ∥ t=\lVert a-y\rVert t = ∥ a − y ∥ c ∈ [ − 1 , 1 ] c\in[-1,1] c ∈ [ − 1 , 1 ] p − a p-a p − a a − y a-y a − y t = 0 t=0 t = 0 ∥ p − y ∥ 2 = 1 + t 2 + 2 t c \lVert p-y\rVert^{2}=1+t^{2}+2tc ∥ p − y ∥ 2 = 1 + t 2 + 2 t c
f ( c ) : = ( 1 + t 2 + 2 t c ) z / 2 − g t c ≤ u + v t z . f(c):=(1+t^{2}+2tc)^{z/2}-gtc\le u+vt^{z}. f ( c ) := ( 1 + t 2 + 2 t c ) z /2 − g t c ≤ u + v t z . At c = 1 c=1 c = 1 c = − 1 c=-1 c = − 1 t t t − t -t − t z ≥ 2 z\ge2 z ≥ 2 f f f c c c [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] c = ± 1 c=\pm1 c = ± 1 1 < z < 2 1<z<2 1 < z < 2 t > 0 t>0 t > 0 f f f f ′ ( c ) = t ( z ℓ ( c ) z − 2 − g ) f'(c)=t(z\ell(c)^{z-2}-g) f ′ ( c ) = t ( z ℓ ( c ) z − 2 − g ) ℓ ( c ) = ( 1 + t 2 + 2 t c ) 1 / 2 \ell(c)=(1+t^{2}+2tc)^{1/2} ℓ ( c ) = ( 1 + t 2 + 2 t c ) 1/2 ∣ 1 − t ∣ \lvert1-t\rvert ∣ 1 − t ∣ 1 + t 1+t 1 + t g ≤ 0 g\le0 g ≤ 0 f f f c = 1 c=1 c = 1 g > 0 g>0 g > 0 ℓ ∗ = ( z / g ) 1 / ( 2 − z ) \ell^{*}=(z/g)^{1/(2-z)} ℓ ∗ = ( z / g ) 1/ ( 2 − z ) ∣ 1 − t ∣ < ℓ ∗ < 1 + t \lvert1-t\rvert<\ell^{*}<1+t ∣ 1 − t ∣ < ℓ ∗ < 1 + t f f f [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] c = ± 1 c=\pm1 c = ± 1 ℓ ( c ) = ℓ ∗ \ell(c)=\ell^{*} ℓ ( c ) = ℓ ∗ g = z ℓ ∗ z − 2 g=z\ell^{*z-2} g = z ℓ ∗ z − 2
ℓ ∗ z − g 2 ( ℓ ∗ 2 − 1 − t 2 ) = ( 1 − z 2 ) ℓ ∗ z + g 2 + g 2 t 2 . \ell^{*z}-\frac{g}{2}(\ell^{*2}-1-t^{2})=\left(1-\frac{z}{2}\right)\ell^{*z}+\frac{g}{2}+\frac{g}{2}t^{2}. ℓ ∗ z − 2 g ( ℓ ∗ 2 − 1 − t 2 ) = ( 1 − 2 z ) ℓ ∗ z + 2 g + 2 g t 2 . As a function of t t t v t z vt^{z} v t z t z − 1 ( g t 2 − z − z v ) t^{z-1}(gt^{2-z}-zv) t z − 1 ( g t 2 − z − z v ) ∣ 1 − ℓ ∗ ∣ ≤ t ≤ 1 + ℓ ∗ \lvert1-\ell^{*}\rvert\le t\le1+\ell^{*} ∣ 1 − ℓ ∗ ∣ ≤ t ≤ 1 + ℓ ∗ t t t ∣ 1 − t ∣ ≤ ℓ ∗ ≤ 1 + t \lvert1-t\rvert\le\ell^{*}\le1+t ∣ 1 − t ∣ ≤ ℓ ∗ ≤ 1 + t c c c 1 1 1 − 1 -1 − 1 t = 0 t=0 t = 0 ℓ ∗ = 1 \ell^{*}=1 ℓ ∗ = 1 g = z g=z g = z 1 ≤ u 1\le u 1 ≤ u f ( c ) ≤ u + v t z f(c)\le u+vt^{z} f ( c ) ≤ u + v t z □ \square □
Lemma 10.3 (Nearly optimal centers). Let z > 1 z>1 z > 1 ( u , g , v ) (u,g,v) ( u , g , v ) z z z α = min { ( z − 1 ) / z , 1 / 2 } \alpha=\min\{(z-1)/z,1/2\} α = min {( z − 1 ) / z , 1/2 } c z c_z c z z z z Q Q Q w w w R d \mathbb{R}^{d} R d a ∈ R d a\in\mathbb{R}^{d} a ∈ R d Ψ Q ( a ) ≤ ( 1 + ϑ ) min c Ψ Q ( c ) \Psi_Q(a)\le(1+\vartheta)\min_c\Psi_Q(c) Ψ Q ( a ) ≤ ( 1 + ϑ ) min c Ψ Q ( c ) ϑ ∈ [ 0 , 1 ] \vartheta\in[0,1] ϑ ∈ [ 0 , 1 ] y 1 , … , y w ∈ R d y_1,\ldots,y_w\in\mathbb{R}^{d} y 1 , … , y w ∈ R d ϕ : Q → { 1 , … , w } \phi:Q\to\{1,\ldots,w\} ϕ : Q → { 1 , … , w }
∑ p ∈ Q ∥ p − y ϕ ( p ) ∥ z ≤ ( u + z − 1 z E ) Ψ Q ( a ) + ( v + E z ) ∑ r = 1 w ∥ a − y r ∥ z , where E = c z ∣ g ∣ α ϑ α . \sum_{p\in Q}\lVert p-y_{\phi(p)}\rVert^{z}\le\left(u+\frac{z-1}{z}E\right)\Psi_Q(a)+\left(v+\frac{E}{z}\right)\sum_{r=1}^{w}\lVert a-y_r\rVert^{z},\qquad\text{where }E=c_z\lvert g\rvert^{\alpha}\vartheta^{\alpha}. p ∈ Q ∑ ∥ p − y ϕ ( p ) ∥ z ≤ ( u + z z − 1 E ) Ψ Q ( a ) + ( v + z E ) r = 1 ∑ w ∥ a − y r ∥ z , where E = c z ∣ g ∣ α ϑ α . In particular, a block whose representative is an optimal center of the block is ( u , v ) (u,v) ( u , v )
Proof. Write A = Ψ Q ( a ) A=\Psi_Q(a) A = Ψ Q ( a ) G = ∇ Ψ Q ( a ) G=\nabla\Psi_Q(a) G = ∇ Ψ Q ( a ) T = ∑ r ∥ a − y r ∥ z T=\sum_r\lVert a-y_r\rVert^{z} T = ∑ r ∥ a − y r ∥ z Q Q Q A = 0 A=0 A = 0 Q Q Q a a a T ≤ v T T\le vT T ≤ v T A > 0 A>0 A > 0 A − min c Ψ Q ( c ) ≤ ϑ min c Ψ Q ( c ) ≤ ϑ A A-\min_c\Psi_Q(c)\le\vartheta\min_c\Psi_Q(c)\le\vartheta A A − min c Ψ Q ( c ) ≤ ϑ min c Ψ Q ( c ) ≤ ϑ A
Ψ Q ( a + h ) ≥ A − ϑ A for all h ∈ R d . (12) \Psi_Q(a+h)\ge A-\vartheta A\qquad\text{for all }h\in\mathbb{R}^{d}. \tag*{(12)} Ψ Q ( a + h ) ≥ A − ϑ A for all h ∈ R d . (12) The cross term. Let ϕ \phi ϕ p p p a a a y ϕ ( p ) y_{\phi(p)} y ϕ ( p ) p ∈ Q p\in Q p ∈ Q p p p Pr [ ϕ ( p ) = r ] = 1 / w \Pr[\phi(p)=r]=1/w Pr [ ϕ ( p ) = r ] = 1/ w p p p r r r ∑ p ∥ p − a ∥ z − 2 ( p − a ) = − G / z \sum_p\lVert p-a\rVert^{z-2}(p-a)=-G/z ∑ p ∥ p − a ∥ z − 2 ( p − a ) = − G / z
E [ ∑ p ∈ Q ∥ p − y ϕ ( p ) ∥ z ] ≤ u A + v T − g z w ⟨ G , ∑ r = 1 w ( a − y r ) ⟩ ≤ u A + v T + ∣ g ∣ z ∥ G ∥ ( T w ) 1 / z , \mathbb{E}\left[\sum_{p\in Q}\lVert p-y_{\phi(p)}\rVert^{z}\right]\le uA+vT-\frac{g}{zw}\left\langle G,\sum_{r=1}^{w}(a-y_r)\right\rangle\le uA+vT+\frac{\lvert g\rvert}{z}\lVert G\rVert\left(\frac{T}{w}\right)^{1/z}, E p ∈ Q ∑ ∥ p − y ϕ ( p ) ∥ z ≤ u A + v T − z w g ⟨ G , r = 1 ∑ w ( a − y r ) ⟩ ≤ u A + v T + z ∣ g ∣ ∥ G ∥ ( w T ) 1/ z , @page 29/48 starts_mid=0 ends_mid=1
Lemma 10.2 (Certificates extend to vectors). Let z > 1 z>1 z > 1 ( u , g , v ) (u,g,v) ( u , g , v ) z z z u ≥ 1 u\ge1 u ≥ 1 v ≥ 1 v\ge1 v ≥ 1 p , a , y ∈ R d p,a,y\in\mathbb{R}^{d} p , a , y ∈ R d
∥ p − y ∥ z ≤ u ∥ p − a ∥ z + g ∥ p − a ∥ z − 2 ⟨ p − a , a − y ⟩ + v ∥ a − y ∥ z . \lVert p-y\rVert^{z}\le u\lVert p-a\rVert^{z}+g\lVert p-a\rVert^{z-2}\langle p-a,a-y\rangle+v\lVert a-y\rVert^{z}. ∥ p − y ∥ z ≤ u ∥ p − a ∥ z + g ∥ p − a ∥ z − 2 ⟨ p − a , a − y ⟩ + v ∥ a − y ∥ z . where the middle term is zero if p = a p=a p = a
Proof. The certificate inequality at t = 0 t=0 t = 0 u ≥ 1 u\ge1 u ≥ 1 ∣ t ∣ z \lvert t\rvert^{z} ∣ t ∣ z t → ∞ t\to\infty t → ∞ v ≥ 1 v\ge1 v ≥ 1 p = a p=a p = a v ≥ 1 v\ge1 v ≥ 1 z z z ∥ p − a ∥ = 1 \lVert p-a\rVert=1 ∥ p − a ∥ = 1 t = ∥ a − y ∥ t=\lVert a-y\rVert t = ∥ a − y ∥ c ∈ [ − 1 , 1 ] c\in[-1,1] c ∈ [ − 1 , 1 ] p − a p-a p − a a − y a-y a − y t = 0 t=0 t = 0 ∥ p − y ∥ 2 = 1 + t 2 + 2 t c \lVert p-y\rVert^{2}=1+t^{2}+2tc ∥ p − y ∥ 2 = 1 + t 2 + 2 t c
f ( c ) : = ( 1 + t 2 + 2 t c ) z / 2 − g t c ≤ u + v t z . f(c):=(1+t^{2}+2tc)^{z/2}-gtc\le u+vt^{z}. f ( c ) := ( 1 + t 2 + 2 t c ) z /2 − g t c ≤ u + v t z . At c = 1 c=1 c = 1 c = − 1 c=-1 c = − 1 t t t − t -t − t z ≥ 2 z\ge2 z ≥ 2 f f f c c c [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] c = ± 1 c=\pm1 c = ± 1 1 < z < 2 1<z<2 1 < z < 2 t > 0 t>0 t > 0 f f f f ′ ( c ) = t ( z ℓ ( c ) z − 2 − g ) f'(c)=t(z\ell(c)^{z-2}-g) f ′ ( c ) = t ( z ℓ ( c ) z − 2 − g ) ℓ ( c ) = ( 1 + t 2 + 2 t c ) 1 / 2 \ell(c)=(1+t^{2}+2tc)^{1/2} ℓ ( c ) = ( 1 + t 2 + 2 t c ) 1/2 ∣ 1 − t ∣ \lvert1-t\rvert ∣ 1 − t ∣ 1 + t 1+t 1 + t g ≤ 0 g\le0 g ≤ 0 f f f c = 1 c=1 c = 1 g > 0 g>0 g > 0 ℓ ∗ = ( z / g ) 1 / ( 2 − z ) \ell^{*}=(z/g)^{1/(2-z)} ℓ ∗ = ( z / g ) 1/ ( 2 − z ) ∣ 1 − t ∣ < ℓ ∗ < 1 + t \lvert1-t\rvert<\ell^{*}<1+t ∣ 1 − t ∣ < ℓ ∗ < 1 + t f f f [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] c = ± 1 c=\pm1 c = ± 1 ℓ ( c ) = ℓ ∗ \ell(c)=\ell^{*} ℓ ( c ) = ℓ ∗ g = z ℓ ∗ z − 2 g=z\ell^{*z-2} g = z ℓ ∗ z − 2
ℓ ∗ z − g 2 ( ℓ ∗ 2 − 1 − t 2 ) = ( 1 − z 2 ) ℓ ∗ z + g 2 + g 2 t 2 . \ell^{*z}-\frac{g}{2}(\ell^{*2}-1-t^{2})=\left(1-\frac{z}{2}\right)\ell^{*z}+\frac{g}{2}+\frac{g}{2}t^{2}. ℓ ∗ z − 2 g ( ℓ ∗ 2 − 1 − t 2 ) = ( 1 − 2 z ) ℓ ∗ z + 2 g + 2 g t 2 . As a function of t t t v t z vt^{z} v t z t z − 1 ( g t 2 − z − z v ) t^{z-1}(gt^{2-z}-zv) t z − 1 ( g t 2 − z − z v ) ∣ 1 − ℓ ∗ ∣ ≤ t ≤ 1 + ℓ ∗ \lvert1-\ell^{*}\rvert\le t\le1+\ell^{*} ∣ 1 − ℓ ∗ ∣ ≤ t ≤ 1 + ℓ ∗ t t t ∣ 1 − t ∣ ≤ ℓ ∗ ≤ 1 + t \lvert1-t\rvert\le\ell^{*}\le1+t ∣ 1 − t ∣ ≤ ℓ ∗ ≤ 1 + t c c c 1 1 1 − 1 -1 − 1 t = 0 t=0 t = 0 ℓ ∗ = 1 \ell^{*}=1 ℓ ∗ = 1 g = z g=z g = z 1 ≤ u 1\le u 1 ≤ u f ( c ) ≤ u + v t z f(c)\le u+vt^{z} f ( c ) ≤ u + v t z □ \square □
Lemma 10.3 (Nearly optimal centers). Let z > 1 z>1 z > 1 ( u , g , v ) (u,g,v) ( u , g , v ) z z z α = min { ( z − 1 ) / z , 1 / 2 } \alpha=\min\{(z-1)/z,1/2\} α = min {( z − 1 ) / z , 1/2 } c z c_z c z z z z Q Q Q w w w R d \mathbb{R}^{d} R d a ∈ R d a\in\mathbb{R}^{d} a ∈ R d Ψ Q ( a ) ≤ ( 1 + ϑ ) min c Ψ Q ( c ) \Psi_Q(a)\le(1+\vartheta)\min_c\Psi_Q(c) Ψ Q ( a ) ≤ ( 1 + ϑ ) min c Ψ Q ( c ) ϑ ∈ [ 0 , 1 ] \vartheta\in[0,1] ϑ ∈ [ 0 , 1 ] y 1 , … , y w ∈ R d y_1,\ldots,y_w\in\mathbb{R}^{d} y 1 , … , y w ∈ R d ϕ : Q → { 1 , … , w } \phi:Q\to\{1,\ldots,w\} ϕ : Q → { 1 , … , w }
∑ p ∈ Q ∥ p − y ϕ ( p ) ∥ z ≤ ( u + z − 1 z E ) Ψ Q ( a ) + ( v + E z ) ∑ r = 1 w ∥ a − y r ∥ z , where E = c z ∣ g ∣ α ϑ α . \sum_{p\in Q}\lVert p-y_{\phi(p)}\rVert^{z}\le\left(u+\frac{z-1}{z}E\right)\Psi_Q(a)+\left(v+\frac{E}{z}\right)\sum_{r=1}^{w}\lVert a-y_r\rVert^{z},\qquad\text{where }E=c_z\lvert g\rvert^{\alpha}\vartheta^{\alpha}. p ∈ Q ∑ ∥ p − y ϕ ( p ) ∥ z ≤ ( u + z z − 1 E ) Ψ Q ( a ) + ( v + z E ) r = 1 ∑ w ∥ a − y r ∥ z , where E = c z ∣ g ∣ α ϑ α . In particular, a block whose representative is an optimal center of the block is ( u , v ) (u,v) ( u , v )
Proof. Write A = Ψ Q ( a ) A=\Psi_Q(a) A = Ψ Q ( a ) G = ∇ Ψ Q ( a ) G=\nabla\Psi_Q(a) G = ∇ Ψ Q ( a ) T = ∑ r ∥ a − y r ∥ z T=\sum_r\lVert a-y_r\rVert^{z} T = ∑ r ∥ a − y r ∥ z Q Q Q A = 0 A=0 A = 0 Q Q Q a a a T ≤ v T T\le vT T ≤ v T A > 0 A>0 A > 0 A − min c Ψ Q ( c ) ≤ ϑ min c Ψ Q ( c ) ≤ ϑ A A-\min_c\Psi_Q(c)\le\vartheta\min_c\Psi_Q(c)\le\vartheta A A − min c Ψ Q ( c ) ≤ ϑ min c Ψ Q ( c ) ≤ ϑ A
Ψ Q ( a + h ) ≥ A − ϑ A for all h ∈ R d . (13) \Psi_Q(a+h)\ge A-\vartheta A\qquad\text{for all }h\in\mathbb{R}^{d}. \tag*{(13)} Ψ Q ( a + h ) ≥ A − ϑ A for all h ∈ R d . (13) The cross term. Let ϕ \phi ϕ p p p a a a y ϕ ( p ) y_{\phi(p)} y ϕ ( p ) p ∈ Q p\in Q p ∈ Q p p p Pr [ ϕ ( p ) = r ] = 1 / w \Pr[\phi(p)=r]=1/w Pr [ ϕ ( p ) = r ] = 1/ w p p p r r r ∑ p ∥ p − a ∥ z − 2 ( p − a ) = − G / z \sum_p\lVert p-a\rVert^{z-2}(p-a)=-G/z ∑ p ∥ p − a ∥ z − 2 ( p − a ) = − G / z
E [ ∑ p ∈ Q ∥ p − y ϕ ( p ) ∥ z ] ≤ u A + v T − g z w ⟨ G , ∑ r = 1 w ( a − y r ) ⟩ ≤ u A + v T + ∣ g ∣ z ∥ G ∥ ( T w ) 1 / z , \mathbb{E}\left[\sum_{p\in Q}\lVert p-y_{\phi(p)}\rVert^{z}\right]\le uA+vT-\frac{g}{zw}\left\langle G,\sum_{r=1}^{w}(a-y_r)\right\rangle\le uA+vT+\frac{\lvert g\rvert}{z}\lVert G\rVert\left(\frac{T}{w}\right)^{1/z}, E p ∈ Q ∑ ∥ p − y ϕ ( p ) ∥ z ≤ u A + v T − z w g ⟨ G , r = 1 ∑ w ( a − y r ) ⟩ ≤ u A + v T + z ∣ g ∣ ∥ G ∥ ( w T ) 1/ z , where the last step uses 1 w ∑ r ∥ a − y r ∥ ≤ ( 1 w ∑ r ∥ a − y r ∥ z ) 1 / z \frac{1}{w}\sum_r \lVert a-y_r\rVert\le\left(\frac{1}{w}\sum_r \lVert a-y_r\rVert^z\right)^{1/z} w 1 ∑ r ∥ a − y r ∥ ≤ ( w 1 ∑ r ∥ a − y r ∥ z ) 1/ z
∥ G ∥ ≤ z c z w 1 / z ϑ α A ( z − 1 ) / z , (14) \lVert G\rVert\le zc_z w^{1/z}\vartheta^\alpha A^{(z-1)/z}, \tag*{(14)} ∥ G ∥ ≤ z c z w 1/ z ϑ α A ( z − 1 ) / z , (14) because then the expected cost is at most u A + v T + E A ( z − 1 ) / z T 1 / z ≤ u A + v T + E ( ( z − 1 ) A + T ) / z uA+vT+EA^{(z-1)/z}T^{1/z}\le uA+vT+E((z-1)A+T)/z u A + v T + E A ( z − 1 ) / z T 1/ z ≤ u A + v T + E (( z − 1 ) A + T ) / z G ≠ 0 G\ne0 G = 0
The gradient for 1 < z ≤ 2 1<z\le2 1 < z ≤ 2 For 1 < z < 2 1<z<2 1 < z < 2 x ↦ ∥ x ∥ z x\mapsto\lVert x\rVert^z x ↦ ∥ x ∥ z 1 < z ≤ 2 1<z\le2 1 < z ≤ 2 ∥ h ∥ z \lVert h\rVert^z ∥ h ∥ z (15) below, which replaces the Hessian bound that we use for z ≥ 2 z\ge2 z ≥ 2 C z = max 0 ≤ s ≤ 1 / 2 ( ( 1 − s ) z − s z + z s z − 1 ) ≤ 3 C_z=\max_{0\le s\le1/2}((1-s)^z-s^z+zs^{z-1})\le3 C z = max 0 ≤ s ≤ 1/2 (( 1 − s ) z − s z + z s z − 1 ) ≤ 3 e , h ∈ R d e,h\in\mathbb{R}^d e , h ∈ R d
∥ e + h ∥ z ≤ ∥ e ∥ z + z ∥ e ∥ z − 2 ⟨ e , h ⟩ + C z ∥ h ∥ z . (15) \lVert e+h\rVert^z\le\lVert e\rVert^z+z\lVert e\rVert^{z-2}\langle e,h\rangle+C_z\lVert h\rVert^z. \tag*{(15)} ∥ e + h ∥ z ≤ ∥ e ∥ z + z ∥ e ∥ z − 2 ⟨ e , h ⟩ + C z ∥ h ∥ z . (15) For h = 0 h=0 h = 0 ∥ h ∥ = 1 \lVert h\rVert=1 ∥ h ∥ = 1 s = ∥ e ∥ s=\lVert e\rVert s = ∥ e ∥ ⟨ e , h ⟩ = s c \langle e,h\rangle=sc ⟨ e , h ⟩ = sc c ∈ [ − 1 , 1 ] c\in[-1,1] c ∈ [ − 1 , 1 ] (15) without the last term is ( s 2 + 2 s c + 1 ) z / 2 − s z − z s z − 1 c (s^2+2sc+1)^{z/2}-s^z-zs^{z-1}c ( s 2 + 2 sc + 1 ) z /2 − s z − z s z − 1 c c c c z s ( s 2 + 2 s c + 1 ) ( z − 2 ) / 2 − s z z − 2 zs(s^2+2sc+1)^{(z-2)/2}-sz^{z-2} z s ( s 2 + 2 sc + 1 ) ( z − 2 ) /2 − s z z − 2 s ≤ 1 / 2 s\le1/2 s ≤ 1/2 c c c c = − 1 c=-1 c = − 1 ( 1 − s ) z − s z + z s z − 1 ≤ C z (1-s)^z-s^z+zs^{z-1}\le C_z ( 1 − s ) z − s z + z s z − 1 ≤ C z s > 1 / 2 s>1/2 s > 1/2 c = − 1 / ( 2 s ) c=-1/(2s) c = − 1/ ( 2 s ) z 2 s z − 2 ≤ z 2 1 − z \frac{z}{2}s^{z-2}\le z2^{1-z} 2 z s z − 2 ≤ z 2 1 − z C z C_z C z s = 1 / 2 s=1/2 s = 1/2 (15) . Applying it to e = p − a e=p-a e = p − a − h -h − h p p p Ψ Q ( a + h ) ≤ A + ⟨ G , h ⟩ + C z w ∥ h ∥ z \Psi_Q(a+h)\le A+\langle G,h\rangle+C_zw\lVert h\rVert^z Ψ Q ( a + h ) ≤ A + ⟨ G , h ⟩ + C z w ∥ h ∥ z h = − τ G / ∥ G ∥ h=-\tau G/\lVert G\rVert h = − τ G / ∥ G ∥ ϑ A ≥ τ ∥ G ∥ − C z w τ z \vartheta A\ge\tau\lVert G\rVert-C_zw\tau^z ϑ A ≥ τ ∥ G ∥ − C z w τ z τ ≥ 0 \tau\ge0 τ ≥ 0 τ \tau τ z − 1 z ∥ G ∥ z / ( z − 1 ) ( z C z w ) − 1 / ( z − 1 ) \frac{z-1}{z}\lVert G\rVert^{z/(z-1)}(zC_zw)^{-1/(z-1)} z z − 1 ∥ G ∥ z / ( z − 1 ) ( z C z w ) − 1/ ( z − 1 ) (14) with c z = C z 1 / z ( z − 1 ) − ( z − 1 ) / z c_z=C_z^{1/z}(z-1)^{-(z-1)/z} c z = C z 1/ z ( z − 1 ) − ( z − 1 ) / z
The gradient for z ≥ 2 z\ge2 z ≥ 2 Let ϱ = ( A / w ) 1 / z \varrho=(A/w)^{1/z} ϱ = ( A / w ) 1/ z ∥ h ∥ ≤ ϱ \lVert h\rVert\le\varrho ∥ h ∥ ≤ ϱ Ψ Q ( a + h ) 1 / z ≤ A 1 / z + w 1 / z ∥ h ∥ ≤ 2 A 1 / z \Psi_Q(a+h)^{1/z}\le A^{1/z}+w^{1/z}\lVert h\rVert\le2A^{1/z} Ψ Q ( a + h ) 1/ z ≤ A 1/ z + w 1/ z ∥ h ∥ ≤ 2 A 1/ z x ↦ ∥ x ∥ z x\mapsto\lVert x\rVert^z x ↦ ∥ x ∥ z z ( z − 1 ) ∥ x ∥ z − 2 z(z-1)\lVert x\rVert^{z-2} z ( z − 1 ) ∥ x ∥ z − 2 Ψ Q \Psi_Q Ψ Q b b b ∥ b − a ∥ ≤ ϱ \lVert b-a\rVert\le\varrho ∥ b − a ∥ ≤ ϱ
z ( z − 1 ) ∑ p ∈ Q ∥ p − b ∥ z − 2 ≤ z ( z − 1 ) w 2 / z Ψ Q ( b ) ( z − 2 ) / z ≤ z ( z − 1 ) 2 z − 2 w 2 / z A ( z − 2 ) / z = : Θ . z(z-1)\sum_{p\in Q}\lVert p-b\rVert^{z-2}\le z(z-1)w^{2/z}\Psi_Q(b)^{(z-2)/z}\le z(z-1)2^{z-2}w^{2/z}A^{(z-2)/z}=:\Theta. z ( z − 1 ) p ∈ Q ∑ ∥ p − b ∥ z − 2 ≤ z ( z − 1 ) w 2/ z Ψ Q ( b ) ( z − 2 ) / z ≤ z ( z − 1 ) 2 z − 2 w 2/ z A ( z − 2 ) / z =: Θ. By Hölder’s inequality, ∥ G ∥ ≤ z ∑ p ∥ p − a ∥ z − 1 ≤ z w 1 / z A ( z − 1 ) / z = z A / ϱ \lVert G\rVert\le z\sum_p\lVert p-a\rVert^{z-1}\le zw^{1/z}A^{(z-1)/z}=zA/\varrho ∥ G ∥ ≤ z ∑ p ∥ p − a ∥ z − 1 ≤ z w 1/ z A ( z − 1 ) / z = z A / ϱ h = − G / Θ h=-G/\Theta h = − G /Θ ∥ h ∥ ≤ ϱ / ( ( z − 1 ) 2 z − 2 ) ≤ ϱ \lVert h\rVert\le\varrho/((z-1)2^{z-2})\le\varrho ∥ h ∥ ≤ ϱ / (( z − 1 ) 2 z − 2 ) ≤ ϱ A − ϑ A ≤ Ψ Q ( a + h ) ≤ A − ∥ G ∥ 2 / Θ + ∥ G ∥ 2 / ( 2 Θ ) A-\vartheta A\le\Psi_Q(a+h)\le A-\lVert G\rVert^2/\Theta+\lVert G\rVert^2/(2\Theta) A − ϑ A ≤ Ψ Q ( a + h ) ≤ A − ∥ G ∥ 2 /Θ + ∥ G ∥ 2 / ( 2Θ ) ∥ G ∥ 2 ≤ 2 Θ ϑ A \lVert G\rVert^2\le2\Theta\vartheta A ∥ G ∥ 2 ≤ 2Θ ϑ A (14) with c z = ( 2 z − 1 ( z − 1 ) / z ) 1 / 2 c_z=(2^{z-1}(z-1)/z)^{1/2} c z = ( 2 z − 1 ( z − 1 ) / z ) 1/2
Candidate centers
The dimension d d d k k k d d d Q Q Q p 0 p_0 p 0 s s s x x x ( s / ( ∥ x − p 0 ∥ + s ) ) z (s/(\lVert x-p_0\rVert+s))^z ( s / (∥ x − p 0 ∥ + s ) ) z s s s p 0 p_0 p 0 z s z − 1 zs^{z-1} z s z − 1 ( ∥ x − p 0 ∥ + s ) z (\lVert x-p_0\rVert+s)^z (∥ x − p 0 ∥ + s ) z Ψ Q \Psi_Q Ψ Q
Lemma 10.4 (Candidate centers). Let P P P n ≥ 2 n\ge2 n ≥ 2 R d \mathbb{R}^d R d z ≥ 1 z\ge1 z ≥ 1 ϑ ∈ ( 0 , 1 ] \vartheta\in(0,1] ϑ ∈ ( 0 , 1 ] N = ⌈ ( z 4 z / ϑ ) 2 ⌉ N=\lceil(z4^z/\vartheta)^2\rceil N = ⌈( z 4 z / ϑ ) 2 ⌉ Γ ⊂ R d \Gamma\subset\mathbb{R}^d Γ ⊂ R d P P P ∣ Γ ∣ ≤ n N + 3 ⌈ log 2 ( 4 n ) ⌉ |\Gamma|\le n^{N+3}\lceil\log_2(4n)\rceil ∣Γ∣ ≤ n N + 3 ⌈ log 2 ( 4 n )⌉ O ( N d ) O(Nd) O ( N d ) Γ \Gamma Γ
min c ∈ Γ Ψ Q ( c ) ≤ ( 1 + ϑ ) min c ∈ R d Ψ Q ( c ) for every nonempty multiset Q ⊆ P . \min_{c\in\Gamma}\Psi_Q(c)\le(1+\vartheta)\min_{c\in\mathbb{R}^d}\Psi_Q(c)\qquad\text{for every nonempty multiset }Q\subseteq P. c ∈ Γ min Ψ Q ( c ) ≤ ( 1 + ϑ ) c ∈ R d min Ψ Q ( c ) for every nonempty multiset Q ⊆ P . Proof. Construction. Γ \Gamma Γ P P P p 0 ∈ P p_0 \in P p 0 ∈ P b b b P P P 0 ≤ j ≤ ⌈ log 2 ( 2 n ) ⌉ 0 \le j \le\lceil\log_2(2n) \rceil 0 ≤ j ≤ ⌈ log 2 ( 2 n )⌉ s = 2 − j b s = 2^{-j}b s = 2 − j b D D D s s s p 0 p_0 p 0 x ∈ P x \in P x ∈ P ℓ x ( c ) = s z ∥ x − c ∥ z / ( ∥ x − p 0 ∥ + s ) z \ell_x(c) = s^z\lVert x-c\rVert^z/(\lVert x-p_0\rVert+s)^z ℓ x ( c ) = s z ∥ x − c ∥ z / (∥ x − p 0 ∥ + s ) z c ∈ D c \in D c ∈ D z s z ∥ x − c ∥ z − 1 / ( ∥ x − p 0 ∥ + s ) z ≤ z s z − 1 zs^z\lVert x-c\rVert^{z-1}/(\lVert x-p_0\rVert+s)^z \le zs^{z-1} z s z ∥ x − c ∥ z − 1 / (∥ x − p 0 ∥ + s ) z ≤ z s z − 1 ( x 1 , … , x N ) ∈ P N (x_1,\ldots,x_N) \in P^N ( x 1 , … , x N ) ∈ P N c 1 = p 0 c_1=p_0 c 1 = p 0 c t + 1 = Π D ( c t − τ G t ) c_{t+1}=\Pi_D(c_t-\tau G_t) c t + 1 = Π D ( c t − τ G t ) t = 1 , … , N t=1,\ldots,N t = 1 , … , N G t G_t G t ℓ x t \ell_{x_t} ℓ x t c t c_t c t c t = x t c_t=x_t c t = x t Π D \Pi_D Π D D D D τ = s 2 − z / ( z N ) \tau=s^{2-z}/(z\sqrt{N}) τ = s 2 − z / ( z N ) c ˉ = 1 N ∑ t = 1 N c t \bar c=\frac{1}{N}\sum_{t=1}^N c_t c ˉ = N 1 ∑ t = 1 N c t Γ \Gamma Γ
Analysis. If all points of Q Q Q Γ \Gamma Γ w = ∣ Q ∣ ≥ 2 w=|Q|\ge2 w = ∣ Q ∣ ≥ 2 c ∗ c^* c ∗ Q Q Q OPT Q = Ψ Q ( c ∗ ) > 0 \operatorname{OPT}_Q=\Psi_Q(c^*)>0 OPT Q = Ψ Q ( c ∗ ) > 0 ϱ = ( OPT Q / w ) 1 / z \varrho=(\operatorname{OPT}_Q/w)^{1/z} ϱ = ( OPT Q / w ) 1/ z p 0 ∈ Q p_0\in Q p 0 ∈ Q c ∗ c^* c ∗ ∥ p 0 − c ∗ ∥ ≤ ϱ \lVert p_0-c^*\rVert\le\varrho ∥ p 0 − c ∗ ∥ ≤ ϱ b b b Q Q Q Q Q Q OPT Q ≤ ( w − 1 ) b z \operatorname{OPT}_Q\le(w-1)b^z OPT Q ≤ ( w − 1 ) b z ϱ ≤ b \varrho\le b ϱ ≤ b b b b 2 1 − z b z 2^{1-z}b^z 2 1 − z b z b ≤ 2 w 1 / z ϱ ≤ 2 n ϱ b\le2w^{1/z}\varrho\le2n\varrho b ≤ 2 w 1/ z ϱ ≤ 2 n ϱ s = 2 − j b s=2^{-j}b s = 2 − j b ϱ ≤ s < 2 ϱ \varrho\le s<2\varrho ϱ ≤ s < 2 ϱ c ∗ ∈ D c^*\in D c ∗ ∈ D Q Q Q x x x ( ∥ x − p 0 ∥ + s ) z / Z (\lVert x-p_0\rVert+s)^z/Z (∥ x − p 0 ∥ + s ) z / Z Z = ∑ x ∈ Q ( ∥ x − p 0 ∥ + s ) z Z=\sum_{x\in Q}(\lVert x-p_0\rVert+s)^z Z = ∑ x ∈ Q (∥ x − p 0 ∥ + s ) z ∥ c ∗ − p 0 ∥ + s ≤ 3 ϱ \lVert c^*-p_0\rVert+s\le3\varrho ∥ c ∗ − p 0 ∥ + s ≤ 3 ϱ
Z 1 / z ≤ Ψ Q ( c ∗ ) 1 / z + w 1 / z ⋅ 3 ϱ = 4 OPT Q 1 / z . Z^{1/z}\le\Psi_Q(c^*)^{1/z}+w^{1/z}\cdot3\varrho=4\operatorname{OPT}_Q^{1/z}. Z 1/ z ≤ Ψ Q ( c ∗ ) 1/ z + w 1/ z ⋅ 3 ϱ = 4 OPT Q 1/ z . For a random x x x ℓ x ( c ) \ell_x(c) ℓ x ( c ) ℓ ˉ ( c ) : = s z Ψ Q ( c ) / Z \bar\ell(c):=s^z\Psi_Q(c)/Z ℓ ˉ ( c ) := s z Ψ Q ( c ) / Z x 1 , … , x N x_1,\ldots,x_N x 1 , … , x N c t c_t c t x t x_t x t G t G_t G t ℓ ˉ \bar\ell ℓ ˉ c t c_t c t D ∋ c ∗ D\ni c^* D ∋ c ∗ c ∗ c^* c ∗
∥ c t + 1 − c ∗ ∥ 2 ≤ ∥ c t − c ∗ ∥ 2 − 2 τ ⟨ G t , c t − c ∗ ⟩ + τ 2 ( z s z − 1 ) 2 = ∥ c t − c ∗ ∥ 2 − 2 τ ⟨ G t , c t − c ∗ ⟩ + s 2 N . \lVert c_{t+1}-c^*\rVert^2\le\lVert c_t-c^*\rVert^2-2\tau\langle G_t,c_t-c^*\rangle+\tau^2(zs^{z-1})^2=\lVert c_t-c^*\rVert^2-2\tau\langle G_t,c_t-c^*\rangle+\frac{s^2}{N}. ∥ c t + 1 − c ∗ ∥ 2 ≤ ∥ c t − c ∗ ∥ 2 − 2 τ ⟨ G t , c t − c ∗ ⟩ + τ 2 ( z s z − 1 ) 2 = ∥ c t − c ∗ ∥ 2 − 2 τ ⟨ G t , c t − c ∗ ⟩ + N s 2 . and by convexity E ⟨ G t , c t − c ∗ ⟩ ≥ E [ ℓ ˉ ( c t ) ] − ℓ ˉ ( c ∗ ) \mathbb{E}\langle G_t,c_t-c^*\rangle\ge\mathbb{E}[\bar\ell(c_t)]-\bar\ell(c^*) E ⟨ G t , c t − c ∗ ⟩ ≥ E [ ℓ ˉ ( c t )] − ℓ ˉ ( c ∗ ) t t t ∥ c 1 − c ∗ ∥ ≤ s \lVert c_1-c^*\rVert\le s ∥ c 1 − c ∗ ∥ ≤ s ℓ ˉ \bar\ell ℓ ˉ
E [ ℓ ˉ ( c ˉ ) ] − ℓ ˉ ( c ∗ ) ≤ 1 N ∑ t = 1 N ( E [ ℓ ˉ ( c t ) ] − ℓ ˉ ( c ∗ ) ) ≤ s 2 + s 2 2 τ N = z s z N . \mathbb{E}[\bar\ell(\bar c)]-\bar\ell(c^*)\le\frac{1}{N}\sum_{t=1}^N\bigl(\mathbb{E}[\bar\ell(c_t)]-\bar\ell(c^*)\bigr)\le\frac{s^2+s^2}{2\tau N}=\frac{zs^z}{\sqrt{N}}. E [ ℓ ˉ ( c ˉ )] − ℓ ˉ ( c ∗ ) ≤ N 1 t = 1 ∑ N ( E [ ℓ ˉ ( c t )] − ℓ ˉ ( c ∗ ) ) ≤ 2 τ N s 2 + s 2 = N z s z . Multiplying by Z / s z ≤ 4 z OPT Q / s z Z/s^z\le4^z\operatorname{OPT}_Q/s^z Z / s z ≤ 4 z OPT Q / s z E [ Ψ Q ( c ˉ ) ] − OPT Q ≤ z 4 z OPT Q / N ≤ ϑ OPT Q \mathbb{E}[\Psi_Q(\bar c)]-\operatorname{OPT}_Q\le z4^z\operatorname{OPT}_Q/\sqrt{N}\le\vartheta\operatorname{OPT}_Q E [ Ψ Q ( c ˉ )] − OPT Q ≤ z 4 z OPT Q / N ≤ ϑ OPT Q Q N ⊆ P N Q^N\subseteq P^N Q N ⊆ P N Γ \Gamma Γ Ψ Q ( c ˉ ) ≤ ( 1 + ϑ ) OPT Q \Psi_Q(\bar c)\le(1+\vartheta)\operatorname{OPT}_Q Ψ Q ( c ˉ ) ≤ ( 1 + ϑ ) OPT Q ∣ Γ ∣ |\Gamma| ∣Γ∣ p 0 p_0 p 0 b b b j j j
Proof of Theorem 1.1
We may assume δ ≤ 1 \delta\le1 δ ≤ 1 OPT > 0 \operatorname{OPT}>0 OPT > 0 OPT = 0 \operatorname{OPT}=0 OPT = 0 ∑ x ⌈ n x / U ⌉ ≤ k \sum_x\lceil n_x/U\rceil\le k ∑ x ⌈ n x / U ⌉ ≤ k n x n_x n x x x x ⌈ n x / U ⌉ \lceil n_x/U\rceil ⌈ n x / U ⌉ x x x 0 0 0 z = 1 z=1 z = 1 ( u , g , v ) = ( 1 , 0 , 1 ) (u,g,v)=(1,0,1) ( u , g , v ) = ( 1 , 0 , 1 ) ( 1 , 1 ) (1,1) ( 1 , 1 ) u + 2 v = 3 = B 1 u+2v=3=\mathcal{B}_1 u + 2 v = 3 = B 1 z > 1 z>1 z > 1 ( u , g , v ) (u,g,v) ( u , g , v ) z z z u + 2 z v ≤ B z + δ / 4 u+2^zv\le\mathcal{B}_z+\delta/4 u + 2 z v ≤ B z + δ /4 z z z δ \delta δ
η = min { 1 10 , δ 32 ⋅ 3 z } . \eta=\min\left\{\frac{1}{10},\frac{\delta}{32\cdot3^z}\right\}. η = min { 10 1 , 32 ⋅ 3 z δ } . let ϑ ≤ η \vartheta\le\eta ϑ ≤ η E = c z ∣ g ∣ ϑ α E=c_z|g|^{\vartheta\alpha} E = c z ∣ g ∣ ϑ α E ( z − 1 + 2 z ) / z ≤ δ / 4 E(z-1+2^z)/z\le\delta/4 E ( z − 1 + 2 z ) / z ≤ δ /4 c z c_z c z α \alpha α z = 1 z=1 z = 1 ϑ = η \vartheta=\eta ϑ = η E = 0 E=0 E = 0 η c = min { 1 / 4 , ϑ / ( 18 z ) } \eta_c=\min\{1/4,\vartheta/(18z)\} η c = min { 1/4 , ϑ / ( 18 z )} ( 1 + 3 η c ) z ≤ 1 + ϑ / 3 (1+3\eta_c)^z\le1+\vartheta/3 ( 1 + 3 η c ) z ≤ 1 + ϑ /3 Locations. Compute the set Γ \Gamma Γ ϑ / 3 \vartheta/3 ϑ /3 Y = Γ Y=\Gamma Y = Γ η c \eta_c η c F = μ ( Γ ) F=\mu(\Gamma) F = μ ( Γ ) F F F F F F κ ′ = κ η c \kappa'=\kappa\eta_c κ ′ = κ η c Q ⊆ P Q\subseteq P Q ⊆ P
min c ∈ F Ψ Q ( c ) ≤ ( 1 + ϑ / 3 ) min c ∈ F Ψ Q ( c ) ≤ ( 1 + ϑ / 3 ) 2 min c ∈ R d Ψ Q ( c ) ≤ ( 1 + ϑ ) min c ∈ R d Ψ Q ( c ) . (16) \min_{c\in F}\Psi_Q(c)\leq(1+\vartheta/3)\min_{c\in F}\Psi_Q(c)\leq(1+\vartheta/3)^2\min_{c\in\mathbb{R}^d}\Psi_Q(c)\leq(1+\vartheta)\min_{c\in\mathbb{R}^d}\Psi_Q(c). \tag*{(16)} c ∈ F min Ψ Q ( c ) ≤ ( 1 + ϑ /3 ) c ∈ F min Ψ Q ( c ) ≤ ( 1 + ϑ /3 ) 2 c ∈ R d min Ψ Q ( c ) ≤ ( 1 + ϑ ) c ∈ R d min Ψ Q ( c ) . (16) Reference solution and blocks. Take an optimal solution, and replace the center of each of its clusters by a point of F F F (16) , this gives a soft solution S ∗ S^* S ∗ k k k F F F cost ( S ∗ ) ≤ ( 1 + ϑ ) OPT \operatorname{cost}(S^*)\leq(1+\vartheta)\operatorname{OPT} cost ( S ∗ ) ≤ ( 1 + ϑ ) OPT Theorem 4.3 , compute an uncapacitated solution with at most k k k F F F ( 1 + η ) (1+\eta) ( 1 + η ) F F F cost ( S ∗ ) \operatorname{cost}(S^*) cost ( S ∗ ) § 2.1 . Then replace the representative of every block P i P_i P i a i ∈ F a_i\in F a i ∈ F Ψ P i \Psi_{P_i} Ψ P i F F F A A A A ≤ ( 1 + η ) cost ( S ∗ ) A\leq(1+\eta)\operatorname{cost}(S^*) A ≤ ( 1 + η ) cost ( S ∗ ) (16) , Ψ P i ( a i ) ≤ ( 1 + ϑ ) min c Ψ P i ( c ) \Psi_{P_i}(a_i)\leq(1+\vartheta)\min_{c}\Psi_{P_i}(c) Ψ P i ( a i ) ≤ ( 1 + ϑ ) min c Ψ P i ( c ) z = 1 z=1 z = 1 ( u ′ , v ′ ) (u',v') ( u ′ , v ′ ) u ′ = u + ( z − 1 ) E / z u'=u+(z-1)E/z u ′ = u + ( z − 1 ) E / z v ′ = v + E / z v'=v+E/z v ′ = v + E / z
β ′ : = u ′ + 2 z v ′ = u + 2 z v + z − 1 + 2 z z E ≤ B z + δ 2 ≤ 2 ⋅ 3 z . \beta' := u'+2^zv'=u+2^zv+\frac{z-1+2^z}{z}E\leq B_z+\frac{\delta}{2}\leq2\cdot3^z. β ′ := u ′ + 2 z v ′ = u + 2 z v + z z − 1 + 2 z E ≤ B z + 2 δ ≤ 2 ⋅ 3 z . The framework. Theorem 2.6 with accuracy η \eta η κ ′ \kappa' κ ′ ( u ′ , v ′ ) (u',v') ( u ′ , v ′ ) S ∗ S^* S ∗ K K K F F F
C 0 ≤ ( β ′ − ( β ′ − 1 ) ε + 4 β ′ η ) cost ( S ∗ ) ≤ ( B z − ( B z − 1 ) ε + δ 2 + δ 4 ) ( 1 + ϑ ) OPT ≤ ( B z − ( B z − 1 ) ε + 7 δ 8 ) OPT , \begin{aligned}
C_0\leq(\beta'-(\beta'-1)\varepsilon+4\beta'\eta)\operatorname{cost}(S^*) \\
&\leq\left(B_z-(B_z-1)\varepsilon+\frac{\delta}{2}+\frac{\delta}{4}\right)(1+\vartheta)\operatorname{OPT}\leq\left(B_z-(B_z-1)\varepsilon+\frac{7\delta}{8}\right)\operatorname{OPT},
\end{aligned} C 0 ≤ ( β ′ − ( β ′ − 1 ) ε + 4 β ′ η ) cost ( S ∗ ) ≤ ( B z − ( B z − 1 ) ε + 2 δ + 4 δ ) ( 1 + ϑ ) OPT ≤ ( B z − ( B z − 1 ) ε + 8 7 δ ) OPT , using β ′ − ( β ′ − 1 ) ε ≤ B z − ( B z − 1 ) ε + ( β ′ − B z ) \beta'-(\beta'-1)\varepsilon\leq B_z-(B_z-1)\varepsilon+(\beta'-B_z) β ′ − ( β ′ − 1 ) ε ≤ B z − ( B z − 1 ) ε + ( β ′ − B z ) 4 β ′ η ≤ 8 ⋅ 3 z η ≤ δ / 4 4\beta'\eta\leq8\cdot3^z\eta\leq\delta/4 4 β ′ η ≤ 8 ⋅ 3 z η ≤ δ /4 ϑ ( B z + 1 ) ≤ 2 ⋅ 3 z η ≤ δ / 16 \vartheta(B_z+1)\leq2\cdot3^z\eta\leq\delta/16 ϑ ( B z + 1 ) ≤ 2 ⋅ 3 z η ≤ δ /16
Distinct centers. Some center of every solution with at most k k k p ≠ p ′ p\ne p' p = p ′ 0 0 0 2 1 − z ∥ p − p ′ ∥ z 2^{1-z}\lVert p-p'\rVert^z 2 1 − z ∥ p − p ′ ∥ z OPT ≥ 2 1 − z a min z \operatorname{OPT}\geq2^{1-z}a_{\min}^z OPT ≥ 2 1 − z a m i n z a min a_{\min} a m i n r 0 > 0 r_0>0 r 0 > 0 ( C 0 1 / z + n 1 / z r 0 ) z = C 0 + δ 16 2 1 − z a min z (C_0^{1/z}+n^{1/z}r_0)^z=C_0+\frac{\delta}{16}2^{1-z}a_{\min}^z ( C 0 1/ z + n 1/ z r 0 ) z = C 0 + 16 δ 2 1 − z a m i n z K K K c c c c + r e 1 c+re_1 c + r e 1 e 1 e_1 e 1 r r r K K K j r 0 / ( K + 1 ) jr_0/(K+1) j r 0 / ( K + 1 ) j = 1 , … , K j=1,\ldots,K j = 1 , … , K c + r e 1 c+re_1 c + r e 1 K K K r 0 r_0 r 0 ( C 0 1 / z + n 1 / z r 0 ) z ≤ C 0 + δ 16 OPT < ( B z − ( B z − 1 ) ε + δ ) OPT (C_0^{1/z}+n^{1/z}r_0)^z\leq C_0+\frac{\delta}{16}\operatorname{OPT}< (B_z-(B_z-1)\varepsilon+\delta)\operatorname{OPT} ( C 0 1/ z + n 1/ z r 0 ) z ≤ C 0 + 16 δ OPT < ( B z − ( B z − 1 ) ε + δ ) OPT
Running time. The set Γ \Gamma Γ n O ( z 2 z 16 κ ′ ) n^{O(z2^{z16}\kappa')} n O ( z 2 z 16 κ ′ ) O ( d ) O(d) O ( d ) O ( d ) O(d) O ( d ) n n n ∣ Γ ∣ |\Gamma| ∣Γ∣ log Δ ( P ∪ F ) \log\Delta(P\cup F) log Δ ( P ∪ F ) κ ′ \kappa' κ ′ z z z ε \varepsilon ε δ \delta δ d d d κ ′ \kappa' κ ′ κ \kappa κ κ \kappa κ z z z ε \varepsilon ε δ \delta δ
Hard Capacities and General Metrics
This section proves Theorems 1.3 and 1.4 . As in § 2.9 , for a finite family S S S F F F cost ( S ) \operatorname{cost}(S) cost ( S ) S S S U U U cost ( S ) = ∞ \operatorname{cost}(S) = \infty cost ( S ) = ∞ ∣ S ∣ U < n |S|U < n ∣ S ∣ U < n S ∗ S^* S ∗ cost ( S ∗ ) \operatorname{cost}(S^*) cost ( S ∗ ) F F F F F F cost ( F ) ≤ O P T \operatorname{cost}(F) \leq\mathrm{OPT} cost ( F ) ≤ OPT O P T \mathrm{OPT} OPT
Projection
For k k k S = F S = F S = F [74 Theorem 1.2] . We allow any family S S S z ≥ 1 z \geq1 z ≥ 1
Lemma 11.1 (Projection). Let S S S F F F cost ( S ) < ∞ \operatorname{cost}(S) < \infty cost ( S ) < ∞ S ∗ S^* S ∗ U U U S ′ ⊆ S S' \subseteq S S ′ ⊆ S ∣ S ′ ∣ ≤ ∣ S ∗ ∣ |S'| \leq|S^*| ∣ S ′ ∣ ≤ ∣ S ∗ ∣ cost ( S ′ ) ≤ ( cost ( S ) 1 / z + 2 cost ( S ∗ ) 1 / z ) z \operatorname{cost}(S') \leq\left(\operatorname{cost}(S)^{1/z} + 2\operatorname{cost}(S^*)^{1/z}\right)^z cost ( S ′ ) ≤ ( cost ( S ) 1/ z + 2 cost ( S ∗ ) 1/ z ) z
Proof. As in § 2.2 , let J J J S ∗ S^* S ∗ j ∈ J j \in J j ∈ J c j c_j c j ℓ j ≤ U \ell_j \leq U ℓ j ≤ U σ \sigma σ S ∗ S^* S ∗ σ ′ \sigma' σ ′ S S S cost ( S ) \operatorname{cost}(S) cost ( S ) U U U π 0 \pi^0 π 0 S × J S \times J S × J π x j 0 = ∣ { p : σ ′ ( p ) = x , σ ( p ) = j } ∣ \pi^0_{xj} = |\{p : \sigma'(p) = x,\ \sigma(p) = j\}| π x j 0 = ∣ { p : σ ′ ( p ) = x , σ ( p ) = j } ∣ ℓ j \ell_j ℓ j U U U x j xj x j dist ( x , c j ) z \operatorname{dist}(x,c_j)^z dist ( x , c j ) z dist ( x , c j ) \operatorname{dist}(x,c_j) dist ( x , c j ) x x x c j c_j c j (6) to the triples ( σ ′ ( p ) , c σ ( p ) , p ) (\sigma'(p),c_{\sigma(p)},p) ( σ ′ ( p ) , c σ ( p ) , p ) π 0 \pi^0 π 0 ( cost ( S ) 1 / z + cost ( S ∗ ) 1 / z ) z \left(\operatorname{cost}(S)^{1/z}+\operatorname{cost}(S^*)^{1/z}\right)^z ( cost ( S ) 1/ z + cost ( S ∗ ) 1/ z ) z π 0 \pi^0 π 0 π \pi π
Call a center x ∈ S x \in S x ∈ S used if its row sum is positive, and full if its row sum is U U U x ≠ x ′ x \ne x' x = x ′ S S S S S S J J J x x x x ′ x' x ′ + 1 +1 + 1 − 1 -1 − 1 − 1 -1 − 1 + 1 +1 + 1 x x x x ′ x' x ′ U U U x x x x ′ x' x ′ ∣ S ∣ ⋅ ∣ J ∣ + ∣ S ∣ |S|\cdot|J| + |S| ∣ S ∣ ⋅ ∣ J ∣ + ∣ S ∣
At the end, every component of the support contains at most one used center of S S S t t t ⌈ t / U ⌉ \lceil t/U\rceil ⌈ t / U ⌉ S S S U U U ⌈ t / U ⌉ \lceil t/U\rceil ⌈ t / U ⌉ J J J S ′ S' S ′ ∣ S ′ ∣ ≤ ∣ J ∣ ≤ ∣ S ∗ ∣ |S'| \leq|J| \leq|S^*| ∣ S ′ ∣ ≤ ∣ J ∣ ≤ ∣ S ∗ ∣ j ∈ J j \in J j ∈ J ℓ j \ell_j ℓ j j j j S ′ S' S ′ U U U (6) to the triples ( p , x , c σ ( p ) ) (p,x,c_{\sigma(p)}) ( p , x , c σ ( p ) ) p p p x x x
( cost ( S ∗ ) 1 / z + ( cost ( S ) 1 / z + cost ( S ∗ ) 1 / z ) ) z = ( cost ( S ) 1 / z + 2 cost ( S ∗ ) 1 / z ) z . \left(\operatorname{cost}(S^*)^{1/z}+\left(\operatorname{cost}(S)^{1/z}+\operatorname{cost}(S^*)^{1/z}\right)\right)^z
=
\left(\operatorname{cost}(S)^{1/z}+2\operatorname{cost}(S^*)^{1/z}\right)^z. ( cost ( S ∗ ) 1/ z + ( cost ( S ) 1/ z + cost ( S ∗ ) 1/ z ) ) z = ( cost ( S ) 1/ z + 2 cost ( S ∗ ) 1/ z ) z . For k k k § 2.8 sharpens the bound.
Lemma 11.2 (Euclidean projection). Let z = 2 z = 2 z = 2 V = R d V = \mathbb{R}^d V = R d S ∗ S^* S ∗ S ′ S' S ′ cost ( S ′ ) ≤ cost ( S ) + 2 cost ( S ∗ ) + 2 cost ( S ) cost ( S ∗ ) \operatorname{cost}(S') \leq\operatorname{cost}(S) + 2\operatorname{cost}(S^*) + 2\sqrt{\operatorname{cost}(S)\operatorname{cost}(S^*)} cost ( S ′ ) ≤ cost ( S ) + 2 cost ( S ∗ ) + 2 cost ( S ) cost ( S ∗ )
Proof. In the last step of the proof of Theorem 11.1, assign the ℓ j \ell_j ℓ j j ∈ J j \in J j ∈ J ℓ j \ell_j ℓ j j j j c j c_j c j ∥ p − x ∥ 2 = ∥ p − c j ∥ 2 + 2 ⟨ p − c j , c j − x ⟩ + ∥ c j − x ∥ 2 \lVert p-x\rVert^2 = \lVert p-c_j\rVert^2 + 2\langle p-c_j,c_j-x\rangle+ \lVert c_j-x\rVert^2 ∥ p − x ∥ 2 = ∥ p − c j ∥ 2 + 2 ⟨ p − c j , c j − x ⟩ + ∥ c j − x ∥ 2 ∑ p : σ ( p ) = j ( p − c j ) = 0 \sum_{p:\sigma(p)=j}(p-c_j)=0 ∑ p : σ ( p ) = j ( p − c j ) = 0 ∑ p : σ ( p ) = j ∥ p − c j ∥ 2 + ∑ x π x j ∥ x − c j ∥ 2 \sum_{p:\sigma(p)=j}\lVert p-c_j\rVert^2 + \sum_x \pi_{xj}\lVert x-c_j\rVert^2 ∑ p : σ ( p ) = j ∥ p − c j ∥ 2 + ∑ x π x j ∥ x − c j ∥ 2 cost ( S ′ ) ≤ cost ( S ∗ ) + cost ( π ) \operatorname{cost}(S') \le\operatorname{cost}(S^*)+\operatorname{cost}(\pi) cost ( S ′ ) ≤ cost ( S ∗ ) + cost ( π ) cost ( π ) ≤ ( cost ( S ) 1 / 2 + cost ( S ∗ ) 1 / 2 ) 2 \operatorname{cost}(\pi)\le(\operatorname{cost}(S)^{1/2}+\operatorname{cost}(S^*)^{1/2})^2 cost ( π ) ≤ ( cost ( S ) 1/2 + cost ( S ∗ ) 1/2 ) 2
Proof of Theorem 1.3 . Let OPT be the optimal hard cost, and let β = 3 z \beta=3^z β = 3 z Theorem 1.2 and β = B z \beta=B_z β = B z R d \mathbb{R}^d R d F F F Theorem 1.2 , every hard solution is a soft solution, so the algorithm of Theorem 1.2 with accuracy δ ′ \delta' δ ′ K K K F F F ( β − ( β − 1 ) ε + δ ′ ) O P T (\beta-(\beta-1)\varepsilon+\delta')\mathrm{OPT} ( β − ( β − 1 ) ε + δ ′ ) OPT R d \mathbb{R}^d R d F ⊂ R d F\subset\mathbb{R}^d F ⊂ R d Theorem 1.1 with accuracy δ ′ \delta' δ ′ K K K R d \mathbb{R}^d R d ( β − ( β − 1 ) ε + δ ′ ) O P T (\beta-(\beta-1)\varepsilon+\delta')\mathrm{OPT} ( β − ( β − 1 ) ε + δ ′ ) OPT U U U
Apply Theorem 11.1 with S = F S=F S = F S ∗ S^* S ∗ w ↦ ( 1 + 2 w 1 / z ) z ( 2 + w − 1 / z ) z − 1 ≤ 2 ⋅ 3 z − 1 w\mapsto(1+2w^{1/z})^z(2+w^{-1/z})^{z-1}\le2\cdot3^{z-1} w ↦ ( 1 + 2 w 1/ z ) z ( 2 + w − 1/ z ) z − 1 ≤ 2 ⋅ 3 z − 1 w ≥ 1 w\ge1 w ≥ 1 δ ′ = δ / ( 2 ⋅ 3 z − 1 ) \delta'=\delta/(2\cdot3^{z-1}) δ ′ = δ / ( 2 ⋅ 3 z − 1 ) cost ( F ) ≤ O P T \operatorname{cost}(F)\le\mathrm{OPT} cost ( F ) ≤ OPT ( 1 + 2 ( β − ( β − 1 ) ε + δ ′ ) 1 / z ) z O P T ≤ ( ( 1 + 2 ( β − ( β − 1 ) ε ) 1 / z ) z + δ ) O P T (1+2(\beta-(\beta-1)\varepsilon+\delta')^{1/z})^z\mathrm{OPT}\le((1+2(\beta-(\beta-1)\varepsilon)^{1/z})^z+\delta)\mathrm{OPT} ( 1 + 2 ( β − ( β − 1 ) ε + δ ′ ) 1/ z ) z OPT ≤ (( 1 + 2 ( β − ( β − 1 ) ε ) 1/ z ) z + δ ) OPT k k k R d \mathbb{R}^d R d Theorem 1.1 to the centroid of its clients, which does not increase the cost, and apply Theorem 11.2 instead. Since δ ′ = δ / 6 \delta'=\delta/6 δ ′ = δ /6 z = 2 z=2 z = 2 ( 1 + 2 ( 5 − 4 ε + δ ′ ) + 2 ( 5 − 4 ε + δ ′ ) 1 / 2 ) O P T ≤ ( 11 − 8 ε + 2 5 − 4 ε + δ ) O P T (1+2(5-4\varepsilon+\delta')+2(5-4\varepsilon+\delta')^{1/2})\mathrm{OPT}\le(11-8\varepsilon+2\sqrt{5-4\varepsilon}+\delta)\mathrm{OPT} ( 1 + 2 ( 5 − 4 ε + δ ′ ) + 2 ( 5 − 4 ε + δ ′ ) 1/2 ) OPT ≤ ( 11 − 8 ε + 2 5 − 4 ε + δ ) OPT
Greedy deletion
We follow the analysis of the reverse greedy algorithm by Chrobak, Kenyon and Young [46 ] . Theorem 11.3 below is the substitutes property of the assignment problem [89 ] , and for hard capacities and z = 1 z=1 z = 1 [8 Section 6] with constant 2. We include short proofs, since we need families with repeated locations and all z ≥ 1 z\ge1 z ≥ 1
Lemma 11.3 (Supermodularity). For finite families S ′ ⊆ S S'\subseteq S S ′ ⊆ S F F F cost ( S ′ ) < ∞ \operatorname{cost}(S')<\infty cost ( S ′ ) < ∞
∑ x ∈ S ∖ S ′ ( cost ( S ∖ { x } ) − cost ( S ) ) ≤ cost ( S ′ ) − cost ( S ) . \sum_{x\in S\setminus S'}\left(\operatorname{cost}(S\setminus\{x\})-\operatorname{cost}(S)\right)\le\operatorname{cost}(S')-\operatorname{cost}(S). x ∈ S ∖ S ′ ∑ ( cost ( S ∖ { x }) − cost ( S ) ) ≤ cost ( S ′ ) − cost ( S ) . Proof. Give every center U U U U U U p p p y y y dist ( p , y ) z \operatorname{dist}(p,y)^z dist ( p , y ) z S ′ ⊆ S ′ ′ ⊆ S S'\subseteq S''\subseteq S S ′ ⊆ S ′′ ⊆ S x ∈ S ′ ′ ∖ S ′ x\in S''\setminus S' x ∈ S ′′ ∖ S ′
cost ( S ′ ′ ) + cost ( S ∖ { x } ) ≤ cost ( S ) + cost ( S ′ ′ ∖ { x } ) , (17) \operatorname{cost}(S'')+\operatorname{cost}(S\setminus\{x\})\le\operatorname{cost}(S)+\operatorname{cost}(S''\setminus\{x\}), \tag*{(17)} cost ( S ′′ ) + cost ( S ∖ { x }) ≤ cost ( S ) + cost ( S ′′ ∖ { x }) , (17) where all four costs are finite, since the families contain S ′ S' S ′ σ 1 \sigma_1 σ 1 σ 2 \sigma_2 σ 2 S S S σ 1 \sigma_1 σ 1 σ 2 \sigma_2 σ 2 x x x S ∖ S ′ ′ S\setminus S'' S ∖ S ′′ σ 1 \sigma_1 σ 1 σ 2 \sigma_2 σ 2 S ′ ′ ∖ { x } S''\setminus\{x\} S ′′ ∖ { x } σ 2 \sigma_2 σ 2 S ∖ S ′ ′ S \setminus S^{\prime\prime} S ∖ S ′′ σ 1 \sigma_1 σ 1 S ′ ∖ { x } S^{\prime} \setminus\{x\} S ′ ∖ { x } σ 2 \sigma_2 σ 2 S ′ ′ S^{\prime\prime} S ′′ σ 1 \sigma_1 σ 1 S ′ ′ S^{\prime\prime} S ′′ σ 2 \sigma_2 σ 2 S ∖ { x } S \setminus\{x\} S ∖ { x } S ′ ′ S^{\prime\prime} S ′′ S ∖ { x } S \setminus\{x\} S ∖ { x } cost ( S ) + cost ( S ′ ′ ∖ { x } ) \operatorname{cost}(S) + \operatorname{cost}(S^{\prime\prime} \setminus\{x\}) cost ( S ) + cost ( S ′′ ∖ { x }) (17) .
Now delete the centers of S ∖ S ′ S \setminus S^{\prime} S ∖ S ′ S S S cost ( S ′ ) − cost ( S ) \operatorname{cost}(S^{\prime}) - \operatorname{cost}(S) cost ( S ′ ) − cost ( S ) (17) , each of them is at least the increase caused by deleting the same center from S S S
One deletion
Lemma 11.4 (One deletion). Let S S S be a family of k + r k+r k + r centers at locations in F F F with r ≥ 1 r \ge1 r ≥ 1 and cost ( S ) < ∞ \operatorname{cost}(S) < \infty cost ( S ) < ∞ and let S ∗ S^* S ∗ be a solution with at most k k k centers at arbitrary points. Some center x ∈ S x \in S x ∈ S satisfies
r ( cost ( S ∖ { x } ) − cost ( S ) ) ≤ ( cost ( S ) 1 / z + 2 cost ( S ∗ ) 1 / z ) z − cost ( S ) . r\bigl(\operatorname{cost}(S \setminus\{x\})-\operatorname{cost}(S)\bigr) \le\bigl(\operatorname{cost}(S)^{1/z}+2\operatorname{cost}(S^*)^{1/z}\bigr)^z-\operatorname{cost}(S). r ( cost ( S ∖ { x }) − cost ( S ) ) ≤ ( cost ( S ) 1/ z + 2 cost ( S ∗ ) 1/ z ) z − cost ( S ) . Proof. By Theorem 11.1, some S ′ ⊆ S S^{\prime} \subseteq S S ′ ⊆ S ∣ S ′ ∣ ≤ k \lvert S^{\prime}\rvert\le k ∣ S ′ ∣ ≤ k cost ( S ′ ) ≤ ( cost ( S ) 1 / z + 2 cost ( S ∗ ) 1 / z ) z \operatorname{cost}(S^{\prime}) \le\bigl(\operatorname{cost}(S)^{1/z}+2\operatorname{cost}(S^*)^{1/z}\bigr)^z cost ( S ′ ) ≤ ( cost ( S ) 1/ z + 2 cost ( S ∗ ) 1/ z ) z S S S S ′ S^{\prime} S ′ cost ( S ′ ) \operatorname{cost}(S^{\prime}) cost ( S ′ ) ∣ S ′ ∣ = k \lvert S^{\prime}\rvert= k ∣ S ′ ∣ = k S ∖ S ′ S \setminus S^{\prime} S ∖ S ′ r r r cost ( S ∖ { x } ) − cost ( S ) \operatorname{cost}(S \setminus\{x\})-\operatorname{cost}(S) cost ( S ∖ { x }) − cost ( S ) cost ( S ′ ) − cost ( S ) \operatorname{cost}(S^{\prime})-\operatorname{cost}(S) cost ( S ′ ) − cost ( S )
For z = 1 z=1 z = 1 2 cost ( S ∗ ) 2\operatorname{cost}(S^*) 2 cost ( S ∗ ) z > 1 z>1 z > 1 cost ( S ) \operatorname{cost}(S) cost ( S ) L z L_z L z
Lemma 11.5 (Greedy deletion). Let ϵ , η ∈ ( 0 , 1 ] \epsilon,\eta\in(0,1] ϵ , η ∈ ( 0 , 1 ] with k ≥ 4 / ( ϵ η ) k \ge4/(\epsilon\eta) k ≥ 4/ ( ϵη ) let ρ ≥ 1 \rho\ge1 ρ ≥ 1 and let S 0 S_0 S 0 be a family of at most 2 k − 1 2k-1 2 k − 1 centers at locations in F F F with cost ( S 0 ) ≤ ρ OPT \operatorname{cost}(S_0) \le\rho\operatorname{OPT} cost ( S 0 ) ≤ ρ OPT where OPT is the optimal soft or the optimal hard cost. Starting from S = S 0 S=S_0 S = S 0 as long as ∣ S ∣ > K \lvert S\rvert> K ∣ S ∣ > K delete a center x ∈ S x \in S x ∈ S that minimizes cost ( S ∖ { x } ) \operatorname{cost}(S \setminus\{x\}) cost ( S ∖ { x }) This takes polynomial time, keeps S ⊆ F S \subseteq F S ⊆ F if S 0 ⊆ F S_0 \subseteq F S 0 ⊆ F and ends with
cost ( S ) ≤ L z ( ρ , ϵ e − η / 2 ) OPT . \operatorname{cost}(S) \le L_z(\rho,\epsilon e^{-\eta/2})\operatorname{OPT}. cost ( S ) ≤ L z ( ρ , ϵ e − η /2 ) OPT . Proof. There are fewer than 2 k 2k 2 k 2 k 2k 2 k ∣ S 0 ∣ ≤ K \lvert S_0\rvert\le K ∣ S 0 ∣ ≤ K b = ⌊ ϵ k ⌋ b=\lfloor\epsilon k\rfloor b = ⌊ ϵ k ⌋ b < ∣ S 0 ∣ − k ≤ k − 1 b<\lvert S_0\rvert-k\le k-1 b < ∣ S 0 ∣ − k ≤ k − 1 b ≥ ϵ k − 1 ≥ 1 b\ge\epsilon k-1\ge1 b ≥ ϵ k − 1 ≥ 1 OPT = 0 \operatorname{OPT}=0 OPT = 0 cost ( S 0 ) = 0 \operatorname{cost}(S_0)=0 cost ( S 0 ) = 0 S ∗ S^* S ∗ OPT > 0 \operatorname{OPT}>0 OPT > 0 ζ r = cost ( S ) / OPT \zeta_r=\operatorname{cost}(S)/\operatorname{OPT} ζ r = cost ( S ) / OPT k + r k+r k + r r = ∣ S 0 ∣ − k , … , b + 1 r=\lvert S_0\rvert-k,\ldots,b+1 r = ∣ S 0 ∣ − k , … , b + 1 S ∗ S^* S ∗
ζ r − 1 ≤ ζ r + ( ζ r 1 / z + 2 ) z − ζ r r . \zeta_{r-1}\le\zeta_r+\frac{(\zeta_r^{1/z}+2)^z-\zeta_r}{r}. ζ r − 1 ≤ ζ r + r ( ζ r 1/ z + 2 ) z − ζ r . The function y ↦ ( y 1 / z + 2 ) z − y y\mapsto(y^{1/z}+2)^z-y y ↦ ( y 1/ z + 2 ) z − y [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) ( 1 + 2 y − 1 / z ) z − 1 − 1 ≥ 0 (1+2y^{-1/z})^{z-1}-1\ge0 ( 1 + 2 y − 1/ z ) z − 1 − 1 ≥ 0 1 / ( ( y 1 / z + 2 ) z − y ) 1/((y^{1/z}+2)^z-y) 1/ (( y 1/ z + 2 ) z − y ) ζ r \zeta_r ζ r ζ r − 1 \zeta_{r-1} ζ r − 1 1 / r 1/r 1/ r ζ r − 1 < ζ r \zeta_{r-1}<\zeta_r ζ r − 1 < ζ r ζ ∣ S 0 ∣ − k ≤ ρ \zeta_{\lvert S_0\rvert-k}\le\rho ζ ∣ S 0 ∣ − k ≤ ρ r r r ρ \rho ρ ζ b \zeta_b ζ b ∑ r = b + 1 k − 1 1 / r ≤ ln ( ( k − 1 ) / b ) \sum_{r=b+1}^{k-1}1/r\le\ln((k-1)/b) ∑ r = b + 1 k − 1 1/ r ≤ ln (( k − 1 ) / b )
ln k − 1 b ≤ ln k ϵ k − 1 = ln 1 ϵ + ln ( 1 + 1 ϵ k − 1 ) ≤ ln 1 ϵ + 1 ϵ k ≤ ln 1 ϵ + η 2 . \ln\frac{k-1}{b}\le\ln\frac{k}{\epsilon k-1}=\ln\frac{1}{\epsilon}+\ln\left(1+\frac{1}{\epsilon k-1}\right)\le\ln\frac{1}{\epsilon}+\frac{1}{\epsilon k}\le\ln\frac{1}{\epsilon}+\frac{\eta}{2}. ln b k − 1 ≤ ln ϵ k − 1 k = ln ϵ 1 + ln ( 1 + ϵ k − 1 1 ) ≤ ln ϵ 1 + ϵ k 1 ≤ ln ϵ 1 + 2 η . By the definition of L z L_z L z ζ b ≤ L z ( ρ , ϵ e − η / 2 ) \zeta_b\le L_z(\rho,\epsilon e^{-\eta/2}) ζ b ≤ L z ( ρ , ϵ e − η /2 )
Proof of Theorem 1.4 . We may assume δ ≤ 1 \delta\le1 δ ≤ 1 η ∈ ( 0 , 1 ] \eta\in(0,1] η ∈ ( 0 , 1 ] η L z ( ( 1 + 2 ρ 1 / z ) z , ϵ ) ≤ δ \eta L_z((1+2\rho^{1/z})^z,\epsilon)\le\delta η L z (( 1 + 2 ρ 1/ z ) z , ϵ ) ≤ δ ϵ \epsilon ϵ ϵ e − η / 2 \epsilon e^{-\eta/2} ϵ e − η /2 L z ( ρ , ϵ ) L_z(\rho,\epsilon) L z ( ρ , ϵ ) L z ( ( 1 + 2 ρ 1 / z ) z , ϵ ) L_z((1+2\rho^{1/z})^z,\epsilon) L z (( 1 + 2 ρ 1/ z ) z , ϵ ) δ \delta δ L z L_z L z ϵ \epsilon ϵ k < 4 / ( ϵ η ) k<4/(\epsilon\eta) k < 4/ ( ϵη ) k k k F F F ( ∣ F ∣ + 1 ) 4 / ( ε η ) (|F|+1)^{4}/(\varepsilon\eta) ( ∣ F ∣ + 1 ) 4 / ( ε η ) A ≤ ρ O P T A \le\rho\mathrm{OPT} A ≤ ρ OPT O P T \mathrm{OPT} OPT § 2.1 gives at most 2 k − 1 2k-1 2 k − 1 A A A S 0 S_0 S 0 cost ( S 0 ) ≤ A ≤ ρ O P T \operatorname{cost}(S_0) \le A \le\rho\mathrm{OPT} cost ( S 0 ) ≤ A ≤ ρ OPT S = F S=F S = F S ∗ S^* S ∗ cost ( F ) ≤ O P T \operatorname{cost}(F) \le\mathrm{OPT} cost ( F ) ≤ OPT S 0 ⊆ F S_0 \subseteq F S 0 ⊆ F ∣ S 0 ∣ ≤ 2 k − 1 |S_0| \le2k-1 ∣ S 0 ∣ ≤ 2 k − 1 cost ( S 0 ) ≤ ( O P T 1 / z + 2 A 1 / z ) z ≤ ( 1 + 2 ρ 1 / z ) z O P T \operatorname{cost}(S_0) \le(\mathrm{OPT}^{1/z}+2A^{1/z})^z \le(1+2\rho^{1/z})^z\mathrm{OPT} cost ( S 0 ) ≤ ( OPT 1/ z + 2 A 1/ z ) z ≤ ( 1 + 2 ρ 1/ z ) z OPT k k k ( 1 + 2 ρ ) (1+2\rho) ( 1 + 2 ρ ) 2 k − 1 2k-1 2 k − 1 [16 ] . By Theorem 11.5, the greedy deletions end with cost at most ( L z ( ρ , ε ) + δ ) O P T (L_z(\rho,\varepsilon)+\delta)\mathrm{OPT} ( L z ( ρ , ε ) + δ ) OPT ( L z ( ( 1 + 2 ρ 1 / z ) z , ε ) + δ ) O P T (L_z((1+2\rho^{1/z})^z,\varepsilon)+\delta)\mathrm{OPT} ( L z (( 1 + 2 ρ 1/ z ) z , ε ) + δ ) OPT
Low dimensions. In the settings of Theorems 1.2 and 1.3 , Theorem 1.4 applies with ρ \rho ρ 1 1 1 P ∪ F P\cup F P ∪ F κ \kappa κ Theorem 4.3 computes such a solution. Otherwise, P P P F F F κ \kappa κ R d \mathbb{R}^d R d F F F η c \eta_c η c Theorem 4.3 with accuracy η c \eta_c η c § 9.1 ; the computed centers are locations in F F F ( 1 + 3 η c ) 2 z ( 1 + η c ) (1+3\eta_c)^{2z}(1+\eta_c) ( 1 + 3 η c ) 2 z ( 1 + η c ) 1 1 1 η c → 0 \eta_c\to0 η c → 0 L z ( ρ , ε ) L_z(\rho,\varepsilon) L z ( ρ , ε ) ρ \rho ρ L z ( 3 z , ε ) + δ L_z(3^z,\varepsilon)+\delta L z ( 3 z , ε ) + δ
Growth of L z L_z L z For s ≥ 0 s\ge0 s ≥ 0 2 z s z − 1 ≤ ( s + 2 ) z − s z ≤ 2 z ( s + 2 ) z − 1 2zs^{z-1}\le(s+2)^z-s^z\le2z(s+2)^{z-1} 2 z s z − 1 ≤ ( s + 2 ) z − s z ≤ 2 z ( s + 2 ) z − 1 y = s z y=s^z y = s z L z L_z L z
∫ ρ 1 / z L z ( ρ , ε ) 1 / z z s z − 1 d s ( s + 2 ) z − s z = ln 1 ε , \int_{\rho^{1/z}}^{L_z(\rho,\varepsilon)^{1/z}} \frac{zs^{z-1}\,ds}{(s+2)^z-s^z}=\ln\frac{1}{\varepsilon}, ∫ ρ 1/ z L z ( ρ , ε ) 1/ z ( s + 2 ) z − s z z s z − 1 d s = ln ε 1 , where the integrand lies between 1 2 ( s s + 2 ) z − 1 \frac{1}{2}\left(\frac{s}{s+2}\right)^{z-1} 2 1 ( s + 2 s ) z − 1 1 2 \frac{1}{2} 2 1 L z ( ρ , ε ) 1 / z ≥ ρ 1 / z + 2 ln ( 1 / ε ) L_z(\rho,\varepsilon)^{1/z}\ge\rho^{1/z}+2\ln(1/\varepsilon) L z ( ρ , ε ) 1/ z ≥ ρ 1/ z + 2 ln ( 1/ ε ) ( s / ( s + 2 ) ) z − 1 → 1 (s/(s+2))^{z-1}\to1 ( s / ( s + 2 ) ) z − 1 → 1 s → ∞ s\to\infty s → ∞ L z ( ρ , ε ) 1 / z = ( 2 + o ( 1 ) ) ln ( 1 / ε ) L_z(\rho,\varepsilon)^{1/z}=(2+o(1))\ln(1/\varepsilon) L z ( ρ , ε ) 1/ z = ( 2 + o ( 1 )) ln ( 1/ ε ) ε → 0 \varepsilon\to0 ε → 0 z z z ρ \rho ρ z = 1 z=1 z = 1 1 2 \frac{1}{2} 2 1 L 1 ( ρ , ε ) = ρ + 2 ln ( 1 / ε ) L_1(\rho,\varepsilon)=\rho+2\ln(1/\varepsilon) L 1 ( ρ , ε ) = ρ + 2 ln ( 1/ ε ) z = 2 z=2 z = 2 1 2 ⋅ s / ( s + 1 ) \frac{1}{2}\cdot s/(s+1) 2 1 ⋅ s / ( s + 1 ) 1 2 ( s − ln ( 1 + s ) ) \frac{1}{2}(s-\ln(1+s)) 2 1 ( s − ln ( 1 + s ))
Open Questions
For soft capacities and in Euclidean space, our ratios start from a constant as ε → 0 \varepsilon\to0 ε → 0 ε \varepsilon ε k k k k k k R 3 \mathbb{R}^3 R 3 ( 1 + ε ) k (1+\varepsilon)k ( 1 + ε ) k ε → 0 \varepsilon\to0 ε → 0 β z \beta_z β z T T T c > 0 c>0 c > 0 k k k 3 − c 3-c 3 − c ( 1 + ε ) k (1+\varepsilon)k ( 1 + ε ) k ε > 0 \varepsilon>0 ε > 0 ( 2 + o ( 1 ) ) z ln z ( 1 / ε ) (2+o(1))^z\ln^z(1/\varepsilon) ( 2 + o ( 1 ) ) z ln z ( 1/ ε ) ε → 0 \varepsilon\to0 ε → 0 ε → 0 \varepsilon\to0 ε → 0 k k k
The Constant B z \mathcal{B}_z B z
This appendix proves Theorem 2.7 and shows how to compute a nearly optimal certificate. All random variables are real and have finite support. Taking expectations in (5) gives weak duality : for every z z z ( u , g , v ) (u,g,v) ( u , g , v ) ξ \xi ξ E [ ξ ] = 0 \mathbb{E}[\xi]=0 E [ ξ ] = 0
E ∣ 1 + ξ ∣ z ≤ u + v E ∣ ξ ∣ z . (18) \mathbb{E}|1+\xi|^z \le u+v\mathbb{E}|\xi|^z. \tag*{(18)} E ∣1 + ξ ∣ z ≤ u + v E ∣ ξ ∣ z . (18) Proof of Theorem 2.7(a). For m ≥ 0 m \ge0 m ≥ 0 V ( m ) V(m) V ( m ) E ∣ 1 + ξ ∣ z \mathbb{E}|1+\xi|^z E ∣1 + ξ ∣ z ξ \xi ξ E [ ξ ] = 0 \mathbb{E}[\xi]=0 E [ ξ ] = 0 E ∣ ξ ∣ z = m \mathbb{E}|\xi|^z=m E ∣ ξ ∣ z = m V ( 2 z ) V(2^z) V ( 2 z ) (18) , every certificate has u + 2 z v ≥ V ( 2 z ) u+2^zv \ge V(2^z) u + 2 z v ≥ V ( 2 z ) E ∣ 1 + ξ ∣ z ≤ ( 1 + ( E ∣ ξ ∣ z ) 1 / z ) z \mathbb{E}|1+\xi|^z \le(1+(\mathbb{E}|\xi|^z)^{1/z})^z E ∣1 + ξ ∣ z ≤ ( 1 + ( E ∣ ξ ∣ z ) 1/ z ) z V V V V ( 0 ) = 1 V(0)=1 V ( 0 ) = 1 V V V ξ 1 \xi_1 ξ 1 ξ 2 \xi_2 ξ 2 m 1 m_1 m 1 m 2 m_2 m 2 s ∈ [ 0 , 1 ] s \in[0,1] s ∈ [ 0 , 1 ] ξ 1 \xi_1 ξ 1 s s s ξ 2 \xi_2 ξ 2 s m 1 + ( 1 − s ) m 2 sm_1+(1-s)m_2 s m 1 + ( 1 − s ) m 2 s E ∣ 1 + ξ 1 ∣ z + ( 1 − s ) E ∣ 1 + ξ 2 ∣ z s\mathbb{E}|1+\xi_1|^z+(1-s)\mathbb{E}|1+\xi_2|^z s E ∣1 + ξ 1 ∣ z + ( 1 − s ) E ∣1 + ξ 2 ∣ z [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) 2 z 2^z 2 z v v v V ( m ) ≤ V ( 2 z ) + v ( m − 2 z ) V(m)\le V(2^z)+v(m-2^z) V ( m ) ≤ V ( 2 z ) + v ( m − 2 z ) m ≥ 0 m\ge0 m ≥ 0 u = V ( 2 z ) − 2 z v u=V(2^z)-2^zv u = V ( 2 z ) − 2 z v h ( t ) = ∣ 1 + t ∣ z − u − v ∣ t ∣ z h(t)=|1+t|^z-u-v|t|^z h ( t ) = ∣1 + t ∣ z − u − v ∣ t ∣ z ξ \xi ξ E [ ξ ] = 0 \mathbb{E}[\xi]=0 E [ ξ ] = 0 E h ( ξ ) ≤ V ( E ∣ ξ ∣ z ) − u − v E ∣ ξ ∣ z ≤ 0 \mathbb{E}h(\xi)\le V(\mathbb{E}|\xi|^z)-u-v\mathbb{E}|\xi|^z\le0 E h ( ξ ) ≤ V ( E ∣ ξ ∣ z ) − u − v E ∣ ξ ∣ z ≤ 0 ξ = 0 \xi=0 ξ = 0 h ( 0 ) ≤ 0 h(0)\le0 h ( 0 ) ≤ 0 t 1 > 0 > t 2 t_1>0>t_2 t 1 > 0 > t 2 t 1 t_1 t 1 t 2 t_2 t 2 − t 2 / ( t 1 − t 2 ) -t_2/(t_1-t_2) − t 2 / ( t 1 − t 2 ) t 1 / ( t 1 − t 2 ) t_1/(t_1-t_2) t 1 / ( t 1 − t 2 ) − t 2 h ( t 1 ) + t 1 h ( t 2 ) ≤ 0 -t_2h(t_1)+t_1h(t_2)\le0 − t 2 h ( t 1 ) + t 1 h ( t 2 ) ≤ 0 h ( t 1 ) / t 1 ≤ h ( t 2 ) / t 2 h(t_1)/t_1\le h(t_2)/t_2 h ( t 1 ) / t 1 ≤ h ( t 2 ) / t 2 g = sup t > 0 h ( t ) / t g=\sup_{t>0}h(t)/t g = sup t > 0 h ( t ) / t h ( t ) ≤ g t h(t)\le gt h ( t ) ≤ g t t t t t > 0 t>0 t > 0 g g g t < 0 t<0 t < 0 g ≤ h ( t ) / t g\le h(t)/t g ≤ h ( t ) / t ( u , g , v ) (u,g,v) ( u , g , v ) u + 2 z v = V ( 2 z ) u+2^zv=V(2^z) u + 2 z v = V ( 2 z ) B z B_z B z
Proof of Theorem 2.7(c). Let u + 2 z v < B z u+2^zv<B_z u + 2 z v < B z ξ \xi ξ E [ ξ ] = 0 \mathbb{E}[\xi]=0 E [ ξ ] = 0 E ∣ ξ ∣ z = 2 z \mathbb{E}|\xi|^z=2^z E ∣ ξ ∣ z = 2 z E ∣ 1 + ξ ∣ z > u + 2 z v \mathbb{E}|1+\xi|^z>u+2^zv E ∣1 + ξ ∣ z > u + 2 z v Pr [ ξ = 0 ] = 0 \Pr[\xi=0]=0 Pr [ ξ = 0 ] = 0 0 0 0 ± s \pm s ± s ξ \xi ξ E ∣ ξ ∣ z = 2 z \mathbb{E}|\xi|^z=2^z E ∣ ξ ∣ z = 2 z E ∣ 1 + ξ ∣ z \mathbb{E}|1+\xi|^z E ∣1 + ξ ∣ z 0 0 0 s s s t t t ξ \xi ξ 1 / t ∈ R 1/t\in\mathbb{R} 1/ t ∈ R ω t = Pr [ ξ = t ] ∣ t ∣ z / 2 z \omega_t=\Pr[\xi=t]|t|^z/2^z ω t = Pr [ ξ = t ] ∣ t ∣ z / 2 z 1 1 1
∑ t ω t ∣ 1 / t ∣ z = 2 − z , ∑ t ω t ∣ 1 / t + 1 ∣ z = 2 − z E ∣ 1 + ξ ∣ z > 2 − z u + v . \sum_t \omega_t|1/t|^z=2^{-z}, \qquad\sum_t\omega_t|1/t+1|^z=2^{-z}\mathbb{E}|1+\xi|^z>2^{-z}u+v. t ∑ ω t ∣1/ t ∣ z = 2 − z , t ∑ ω t ∣1/ t + 1 ∣ z = 2 − z E ∣1 + ξ ∣ z > 2 − z u + v . The point 0 0 0 z > 1 z>1 z > 1 c ↦ ∑ t ω t ∣ 1 / t − c ∣ z c\mapsto\sum_t\omega_t|1/t-c|^z c ↦ ∑ t ω t ∣1/ t − c ∣ z c = 0 c=0 c = 0 − z ∑ t ω t ∣ 1 / t ∣ z − 2 / t = − z 2 − z E [ ξ ] = 0 -z\sum_t\omega_t|1/t|^{z-2}/t=-z2^{-z}\mathbb{E}[\xi]=0 − z ∑ t ω t ∣1/ t ∣ z − 2 / t = − z 2 − z E [ ξ ] = 0 z = 1 z=1 z = 1 1 / t 1/t 1/ t t > 0 t>0 t > 0 E [ max { ξ , 0 } ] / 2 = 1 4 E ∣ ξ ∣ = 1 2 \mathbb{E}[\max\{\xi,0\}]/2=\frac14\mathbb{E}|\xi|=\frac12 E [ max { ξ , 0 }] /2 = 4 1 E ∣ ξ ∣ = 2 1 0 0 0 1 / w 1/w 1/ w Q Q Q w w w 0 0 0 Q Q Q z = 1 z=1 z = 1 w / 2 w/2 w /2 0 0 0 z > 1 z>1 z > 1 0 0 0 w w w ∑ p ∈ Q ∣ p + 1 ∣ z > u ∑ p ∈ Q ∣ p ∣ z + v w \sum_{p\in Q}|p+1|^z>u\sum_{p\in Q}|p|^z+vw ∑ p ∈ Q ∣ p + 1 ∣ z > u ∑ p ∈ Q ∣ p ∣ z + v w 0 0 0 Q Q Q w w w − 1 -1 − 1 ∑ p ∈ Q ∣ p + 1 ∣ z \sum_{p\in Q}|p+1|^z ∑ p ∈ Q ∣ p + 1 ∣ z u u u ∑ p ∈ Q ∣ p ∣ z \sum_{p\in Q}|p|^z ∑ p ∈ Q ∣ p ∣ z v v v w w w Q Q Q ( u , v ) (u,v) ( u , v )
Proof of Theorem 2.7(b). B z < 3 z B_z<3^z B z < 3 z z > 1 z>1 z > 1 ( 3 z − 1 , 0 , ( 3 / 2 ) z − 1 ) (3^{z-1},0,(3/2)^{z-1}) ( 3 z − 1 , 0 , ( 3/2 ) z − 1 ) h 0 ( t ) = ∣ 1 + t ∣ z − 3 z − 1 − ( 3 / 2 ) z − 1 ∣ t ∣ z ≤ 0 h_0(t)=|1+t|^z-3^{z-1}-(3/2)^{z-1}|t|^z\le0 h 0 ( t ) = ∣1 + t ∣ z − 3 z − 1 − ( 3/2 ) z − 1 ∣ t ∣ z ≤ 0 ∣ 1 + t ∣ = 1 + ∣ t ∣ |1+t|=1+|t| ∣1 + t ∣ = 1 + ∣ t ∣ t ≥ 0 t\ge0 t ≥ 0 3 = 3 2 ∣ t ∣ 3=\frac32|t| 3 = 2 3 ∣ t ∣ h 0 ( t ) < 0 h_0(t)<0 h 0 ( t ) < 0 t ≠ 2 t\ne2 t = 2 t → − ∞ t\to-\infty t → − ∞ − h 0 ( t ) -h_0(t) − h 0 ( t ) ( ( 3 / 2 ) z − 1 − 1 ) ∣ t ∣ z ((3/2)^{z-1}-1)|t|^z (( 3/2 ) z − 1 − 1 ) ∣ t ∣ z − h 0 ( 1 ) > 0 -h_0(1)>0 − h 0 ( 1 ) > 0 − h 0 ( t ) / ( 1 − t ) -h_0(t)/(1-t) − h 0 ( t ) / ( 1 − t ) ( − ∞ , 1 ) (-\infty,1) ( − ∞ , 1 ) ∞ \infty ∞ g > 0 g>0 g > 0 ( 3 z − 1 − g , g , ( 3 / 2 ) z − 1 ) (3^{z-1}-g,g,(3/2)^{z-1}) ( 3 z − 1 − g , g , ( 3/2 ) z − 1 ) h 0 ( t ) ≤ g ( t − 1 ) h_0(t)\le g(t-1) h 0 ( t ) ≤ g ( t − 1 ) t t t 3 z − g 3^z-g 3 z − g
z = 1 z=1 z = 1 ( 1 , 0 , 1 ) (1,0,1) ( 1 , 0 , 1 ) B 1 ≤ 3 B_1\le3 B 1 ≤ 3 t = 0 t=0 t = 0 u ≥ 1 u\ge1 u ≥ 1 ∣ t ∣ |t| ∣ t ∣ t → ∞ t\to\infty t → ∞ t → − ∞ t\to-\infty t → − ∞ v + g ≥ 1 v+g\ge1 v + g ≥ 1 v − g ≥ 1 v-g\ge1 v − g ≥ 1 u + 2 v ≥ 3 u+2v\ge3 u + 2 v ≥ 3
z = 2 z=2 z = 2 ( 1 , 2 , 1 ) (1,2,1) ( 1 , 2 , 1 ) B 2 ≤ 5 B_2\le5 B 2 ≤ 5 u , v ≥ 1 u,v\ge1 u , v ≥ 1 B 2 ≥ 5 B_2\ge5 B 2 ≥ 5
z = 4 z=4 z = 4 q > 0 q>0 q > 0 a q = 3 / q + 2 / q 2 a_q=3/q+2/q^2 a q = 3/ q + 2/ q 2 t ∈ R t\in\mathbb{R} t ∈ R
( 1 + q a q 2 ) + ( 4 + 4 a q ) t + ( 1 + q ) t 4 − ( 1 + t ) 4 = q ( t 2 − 2 t q − a q ) 2 ≥ 0 , (1+qa_q^2)+(4+4a_q)t+(1+q)t^4-(1+t)^4=q\left(t^2-\frac{2t}{q}-a_q\right)^2\ge0, ( 1 + q a q 2 ) + ( 4 + 4 a q ) t + ( 1 + q ) t 4 − ( 1 + t ) 4 = q ( t 2 − q 2 t − a q ) 2 ≥ 0 , so ( 1 + q a q 2 , 4 + 4 a q , 1 + q ) (1+qa_q^2,\,4+4a_q,\,1+q) ( 1 + q a q 2 , 4 + 4 a q , 1 + q ) f ( q ) = 17 + 16 q + 9 / q + 12 / q 2 + 4 / q 3 f(q)=17+16q+9/q+12/q^2+4/q^3 f ( q ) = 17 + 16 q + 9/ q + 12/ q 2 + 4/ q 3 q ∗ q^* q ∗ 16 q 4 − 9 q 2 − 24 q − 12 16q^4-9q^2-24q-12 16 q 4 − 9 q 2 − 24 q − 12 t 2 − 2 t / q ∗ − a q ∗ t^2-2t/q^*-a_{q^*} t 2 − 2 t / q ∗ − a q ∗ r > 0 > − s r>0>-s r > 0 > − s r − s = 2 / q ∗ r-s=2/q^* r − s = 2/ q ∗ r s = a q ∗ rs=a_{q^*} r s = a q ∗ ξ \xi ξ r r r − s -s − s s / ( r + s ) s/(r+s) s / ( r + s ) r / ( r + s ) r/(r+s) r / ( r + s ) E ξ 4 = r s ( r 2 − r s + s 2 ) = a q ∗ ( a q ∗ + 4 / q ∗ 2 ) = 9 / q ∗ 2 + 24 / q ∗ 3 + 12 / q ∗ 4 = 16 \mathbb{E}\xi^4=rs(r^2-rs+s^2)=a_{q^*}(a_{q^*}+4/q^{*2})=9/q^{*2}+24/q^{*3}+12/q^{*4}=16 E ξ 4 = r s ( r 2 − r s + s 2 ) = a q ∗ ( a q ∗ + 4/ q ∗ 2 ) = 9/ q ∗ 2 + 24/ q ∗ 3 + 12/ q ∗ 4 = 16 r r r − s -s − s E ( 1 + ξ ) 4 = f ( q ∗ ) \mathbb{E}(1+\xi)^4=f(q^*) E ( 1 + ξ ) 4 = f ( q ∗ ) B 4 = f ( q ∗ ) ≈ 53.4056 B_4=f(q^*)\approx53.4056 B 4 = f ( q ∗ ) ≈ 53.4056
z = 3 / 2 z=3/2 z = 3/2 z = 3 z=3 z = 3
( 1107 1000 , 2809 4000 , 4289 4000 ) for z = 3 2 , ( 21263 5000 , 5767 1000 , 15723 10000 ) for z = 3 \left(\frac{1107}{1000},\frac{2809}{4000},\frac{4289}{4000}\right)\quad\text{for }z=\frac{3}{2},\qquad
\left(\frac{21263}{5000},\frac{5767}{1000},\frac{15723}{10000}\right)\quad\text{for }z=3 ( 1000 1107 , 4000 2809 , 4000 4289 ) for z = 2 3 , ( 5000 21263 , 1000 5767 , 10000 15723 ) for z = 3 are certificates, with values 1107 1000 + 4289 2000 2 < 4.1398 \frac{1107}{1000}+\frac{4289}{2000}\sqrt{2}<4.1398 1000 1107 + 2000 4289 2 < 4.1398 16.831 16.831 16.831 z = 3 z=3 z = 3 t ≥ 0 t\ge0 t ≥ 0 − 1 ≤ t ≤ 0 -1\le t\le0 − 1 ≤ t ≤ 0 t ≤ − 1 t\le-1 t ≤ − 1 z = 3 / 2 z=3/2 z = 3/2 t = x 2 t=x^2 t = x 2 t ≥ 0 t\ge0 t ≥ 0 t = − x 2 t=-x^2 t = − x 2 t ≤ 0 t\le0 t ≤ 0 u ± g x 2 + v x 3 u\pm gx^2+vx^3 u ± g x 2 + v x 3 x ≥ 0 x\ge0 x ≥ 0 [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) [ 0 , 1 ] [0,1] [ 0 , 1 ] [ 1 , ∞ ) [1,\infty) [ 1 , ∞ ) 0.4954 0.4954 0.4954 − 26.15 -26.15 − 26.15 z = 3 / 2 z=3/2 z = 3/2 2.9486 2.9486 2.9486 − 1.1093 -1.1093 − 1.1093 z = 3 z=3 z = 3 E ∣ ξ ∣ z = 2 z \mathbb{E}|\xi|^z=2^z E ∣ ξ ∣ z = 2 z E ∣ 1 + ξ ∣ z \mathbb{E}|1+\xi|^z E ∣1 + ξ ∣ z 4.1396 4.1396 4.1396 16.8308 16.8308 16.8308
Computing a certificate. Theorem 1.1 needs a certificate whose value is close to B z B_z B z
Lemma A.1. For fixed z > 1 z>1 z > 1 γ ′ ∈ ( 0 , 1 ] \gamma'\in(0,1] γ ′ ∈ ( 0 , 1 ] z z z ( u , g , v ) (u,g,v) ( u , g , v ) u + 2 z v ≤ B z + γ ′ u+2^zv\le B_z+\gamma' u + 2 z v ≤ B z + γ ′ z z z γ ′ \gamma' γ ′
Proof. Let ( u ∗ , g ∗ , v ∗ ) (u^*,g^*,v^*) ( u ∗ , g ∗ , v ∗ ) u ∗ + 2 z v ∗ ≤ 3 z u^*+2^zv^*\le3^z u ∗ + 2 z v ∗ ≤ 3 z u ∗ , v ∗ ≥ 1 u^*,v^*\ge1 u ∗ , v ∗ ≥ 1 u ∗ , v ∗ ≤ 3 z u^*,v^*\le3^z u ∗ , v ∗ ≤ 3 z t = − 1 t=-1 t = − 1 g ∗ ≤ u ∗ + v ∗ ≤ 2 ⋅ 3 z g^*\le u^*+v^*\le2\cdot3^z g ∗ ≤ u ∗ + v ∗ ≤ 2 ⋅ 3 z g ∗ ≥ 0 g^*\ge0 g ∗ ≥ 0 g ∗ < 0 g^*<0 g ∗ < 0 ( u ∗ , 0 , v ∗ ) (u^*,0,v^*) ( u ∗ , 0 , v ∗ ) ∣ 1 − t ∣ z ≤ ( 1 + t ) z ≤ u ∗ + g ∗ t + v ∗ t z |1-t|^z\le(1+t)^z\le u^*+g^*t+v^*t^z ∣1 − t ∣ z ≤ ( 1 + t ) z ≤ u ∗ + g ∗ t + v ∗ t z t ≥ 0 t\ge0 t ≥ 0 ν = γ ′ / 2 z + 2 \nu=\gamma'/2^{z+2} ν = γ ′ / 2 z + 2
t 0 = max { 1 , z 2 z + 1 ν , ( 4 ⋅ 3 z ν ) 1 / ( z − 1 ) } . t_0=\max\left\{1,\frac{z2^{z+1}}{\nu},\left(\frac{4\cdot3^z}{\nu}\right)^{1/(z-1)}\right\}. t 0 = max { 1 , ν z 2 z + 1 , ( ν 4 ⋅ 3 z ) 1/ ( z − 1 ) } . For ∣ t ∣ ≥ t 0 |t|\ge t_0 ∣ t ∣ ≥ t 0 ∣ 1 + t ∣ z ≤ ∣ t ∣ z + z 2 z − 1 ∣ t ∣ z − 1 ≤ ∣ t ∣ z + ν 2 ∣ t ∣ z |1+t|^z\le|t|^z+z2^{z-1}|t|^{z-1}\le|t|^z+\frac{\nu}{2}|t|^z ∣1 + t ∣ z ≤ ∣ t ∣ z + z 2 z − 1 ∣ t ∣ z − 1 ≤ ∣ t ∣ z + 2 ν ∣ t ∣ z 2 ⋅ 3 z ∣ t ∣ ≤ ν 2 ∣ t ∣ z 2\cdot3^z|t|\le\frac{\nu}{2}|t|^z 2 ⋅ 3 z ∣ t ∣ ≤ 2 ν ∣ t ∣ z u ≥ 1 u\ge1 u ≥ 1 ∣ g ∣ ≤ 2 ⋅ 3 z |g|\le2\cdot3^z ∣ g ∣ ≤ 2 ⋅ 3 z v ≥ 1 + ν v\ge1+\nu v ≥ 1 + ν ∣ t ∣ ≥ t 0 |t|\ge t_0 ∣ t ∣ ≥ t 0 [ − t 0 , t 0 ] [-t_0,t_0] [ − t 0 , t 0 ] t ↦ ∣ 1 + t ∣ z − g t − v ∣ t ∣ z t\mapsto|1+t|^z-gt-v|t|^z t ↦ ∣1 + t ∣ z − g t − v ∣ t ∣ z Θ = z ( 1 + t 0 ) z − 1 + 2 ⋅ 3 z + ( 3 z + 1 ) z t 0 z − 1 \Theta=z(1+t_0)^{z-1}+2\cdot3^z+(3^z+1)zt_0^{z-1} Θ = z ( 1 + t 0 ) z − 1 + 2 ⋅ 3 z + ( 3 z + 1 ) z t 0 z − 1 ∣ g ∣ ≤ 2 ⋅ 3 z |g|\le2\cdot3^z ∣ g ∣ ≤ 2 ⋅ 3 z v ≤ 3 z + 1 v\le3^z+1 v ≤ 3 z + 1 Ω \Omega Ω [ − t 0 , t 0 ] [-t_0,t_0] [ − t 0 , t 0 ] γ ′ / ( 4 Θ ) \gamma'/(4\Theta) γ ′ / ( 4Θ ) u + 2 z v u+2^zv u + 2 z v 1 ≤ u ≤ 3 z 1\le u\le3^z 1 ≤ u ≤ 3 z 0 ≤ g ≤ 2 ⋅ 3 z 0\le g\le2\cdot3^z 0 ≤ g ≤ 2 ⋅ 3 z 1 + ν ≤ v ≤ 3 z + ν 1+\nu\le v\le3^z+\nu 1 + ν ≤ v ≤ 3 z + ν t ∈ Ω t\in\Omega t ∈ Ω ( u ∗ , g ∗ , v ∗ + ν ) (u^*,g^*,v^*+\nu) ( u ∗ , g ∗ , v ∗ + ν ) B z + 2 z ν ≤ B z + γ ′ / 4 B_z+2^z\nu\le B_z+\gamma'/4 B z + 2 z ν ≤ B z + γ ′ /4 ( u , g , v ) (u,g,v) ( u , g , v ) t ∈ [ − t 0 , t 0 ] t\in[-t_0,t_0] t ∈ [ − t 0 , t 0 ] γ ′ / ( 4 Θ ) \gamma'/(4\Theta) γ ′ / ( 4Θ ) Ω \Omega Ω ( u + γ ′ / 4 , g , v ) (u+\gamma'/4,g,v) ( u + γ ′ /4 , g , v ) B z + γ ′ / 2 B_z+\gamma'/2 B z + γ ′ /2 z z z γ ′ \gamma' γ ′ ∣ 1 + t ∣ z |1+t|^z ∣1 + t ∣ z ∣ t ∣ z |t|^z ∣ t ∣ z γ ′ / ( 16 ⋅ 3 z ) \gamma'/(16\cdot3^z) γ ′ / ( 16 ⋅ 3 z ) Ω \Omega Ω v ≥ 0 v\ge0 v ≥ 0 ( u ∗ + γ ′ / 8 , g ∗ , v ∗ + ν ) (u^*+\gamma'/8,g^*,v^*+\nu) ( u ∗ + γ ′ /8 , g ∗ , v ∗ + ν ) ( 1 + v ∗ + ν ) γ ′ / ( 16 ⋅ 3 z ) ≤ γ ′ / 8 (1+v^*+\nu)\gamma'/(16\cdot3^z)\le\gamma'/8 ( 1 + v ∗ + ν ) γ ′ / ( 16 ⋅ 3 z ) ≤ γ ′ /8 B z + 5 γ ′ / 8 B_z+5\gamma'/8 B z + 5 γ ′ /8
Implementation with Rational Numbers
In this appendix, the input consists of rational numbers: the coordinates of the clients in the Euclidean setting, or the matrix of distances between the points of P ∪ F P \cup F P ∪ F z z z w ↦ w z w \mapsto w^z w ↦ w z η \eta η 1 + O ( η ) 1 + O(\eta) 1 + O ( η ) O ( η ) ⋅ O P T O(\eta) \cdot\mathrm{OPT} O ( η ) ⋅ OPT O O O O P T ‾ \underline{\mathrm{OPT}} OPT O P T \mathrm{OPT} OPT
A lower bound. Whether O P T = 0 \mathrm{OPT}=0 OPT = 0 ⌈ n x / U ⌉ \lceil n_x/U\rceil ⌈ n x / U ⌉ x x x n x n_x n x n x ≤ U n_x\le U n x ≤ U O P T = 0 \mathrm{OPT}=0 OPT = 0 O P T ≥ O P T ‾ : = 2 1 − z a min z \mathrm{OPT}\ge\underline{\mathrm{OPT}}:=2^{1-z}a_{\min}^{z} OPT ≥ OPT := 2 1 − z a m i n z § 10.3 , where a min a_{\min} a m i n O P T ≥ O P T ‾ \mathrm{OPT}\ge\underline{\mathrm{OPT}} OPT ≥ OPT z z z
Working metric. Distances may be irrational, for instance Euclidean distances between rational points. Let η w \eta_w η w η \eta η η c \eta_c η c x , y x,y x , y ℓ x y = ℓ y x \ell_{xy}=\ell_{yx} ℓ x y = ℓ y x dist ( x , y ) ≤ ℓ x y ≤ ( 1 + η w / z ) dist ( x , y ) \operatorname{dist}(x,y)\le\ell_{xy}\le(1+\eta_w/z)\operatorname{dist}(x,y) dist ( x , y ) ≤ ℓ x y ≤ ( 1 + η w / z ) dist ( x , y ) d ˉ \bar d d ˉ x x x y y y dist ( x , y ) \operatorname{dist}(x,y) dist ( x , y ) ( 1 + η w / z ) dist ( x , y ) (1+\eta_w/z)\operatorname{dist}(x,y) ( 1 + η w / z ) dist ( x , y ) dist ≤ d ˉ ≤ ( 1 + η w / z ) dist \operatorname{dist}\le\bar d\le(1+\eta_w/z)\operatorname{dist} dist ≤ d ˉ ≤ ( 1 + η w / z ) dist κ \kappa κ dist \operatorname{dist} dist 2 κ 2\kappa 2 κ d ˉ \bar d d ˉ d ˉ \bar d d ˉ r r r dist \operatorname{dist} dist r r r 2 2 κ 2^{2\kappa} 2 2 κ dist \operatorname{dist} dist r / 4 r/4 r /4 d ˉ \bar d d ˉ ( 1 + η w / z ) r / 4 ≤ r / 2 (1+\eta_w/z)r/4\le r/2 ( 1 + η w / z ) r /4 ≤ r /2 d ˉ \bar d d ˉ κ \kappa κ 2 κ 2\kappa 2 κ Λ = 2 14 κ / γ \Lambda=2^{14\kappa/\gamma} Λ = 2 14 κ / γ M = ⌈ ( 4 / λ ) 2 κ ⌉ M=\lceil(4/\lambda)^{2\kappa}\rceil M = ⌈( 4/ λ ) 2 κ ⌉ (8) hold for d ˉ \bar d d ˉ d ˉ ( x , y ) z ≥ dist ( x , y ) z \bar d(x,y)^z\ge\operatorname{dist}(x,y)^z d ˉ ( x , y ) z ≥ dist ( x , y ) z ( ( 1 + η / z ) d ( x , y ) ) z ≤ e η z dist ( x , y ) z ((1+\eta/z)d(x,y))^z\le e^{\eta z}\operatorname{dist}(x,y)^z (( 1 + η / z ) d ( x , y ) ) z ≤ e η z dist ( x , y ) z Λ \Lambda Λ e η dist ( x , y ) z e^\eta\operatorname{dist}(x,y)^z e η dist ( x , y ) z dist \operatorname{dist} dist ( 1 + 3 η c ) ( 1 + η c / z ) (1+3\eta_c)(1+\eta_c/z) ( 1 + 3 η c ) ( 1 + η c / z ) 1 + 3 η c 1+3\eta_c 1 + 3 η c
Costs. The dynamic program uses rational upper approximations of the arc costs with additive error at most η O P T / ( 4 n max { 1 , ν } ) \eta\mathrm{OPT}/(4n\max\{1,\nu\}) η OPT / ( 4 n max { 1 , ν }) ( u − 1 ) A i (u-1)A_i ( u − 1 ) A i η O P T / ( 4 n ) \eta\mathrm{OPT}/(4n) η OPT / ( 4 n ) 0 0 0 n n n m ≤ n m\le n m ≤ n η O P T / 2 \eta\mathrm{OPT}/2 η OPT /2 η O P T / 2 \eta\mathrm{OPT}/2 η OPT /2 dist ( x , y ) z \operatorname{dist}(x,y)^z dist ( x , y ) z η O P T / ( 4 n ) \eta\mathrm{OPT}/(4n) η OPT / ( 4 n ) n n n η O P T / 4 \eta\mathrm{OPT}/4 η OPT /4 Theorem 4.3 is run with costs approximated to relative error η / ( 4 k ) \eta/(4k) η / ( 4 k ) 1 − η / ( 4 k ) 1-\eta/(4k) 1 − η / ( 4 k ) 1 + O ( η ) 1+O(\eta) 1 + O ( η ) k / η k/\eta k / η O P T \mathrm{OPT} OPT
Candidate centers. The construction of Theorem 10.4 uses square roots and real powers. We run it with N N N N ′ = 100 N N'=100N N ′ = 100 N s s s s ~ \tilde{s} s ~ s ≤ s ~ ≤ ( 1 + 1 / ( 8 z ) ) s s\leq\tilde{s}\leq(1+1/(8z))s s ≤ s ~ ≤ ( 1 + 1/ ( 8 z )) s s ~ z z − 1 / N ′ \tilde{s}z^{z-1}/\sqrt{N'} s ~ z z − 1 / N ′ s ~ / ( z N ′ ) \tilde{s}/(zN') s ~ / ( z N ′ ) τ = s ~ 2 − z / ( z N ′ ) \tau=\tilde{s}^{2-z}/(z\sqrt{N'}) τ = s ~ 2 − z / ( z N ′ ) ∥ c ∗ − p 0 ∥ + s ~ ≤ ( 3 + 1 / 4 z ) ϱ \lVert c^*-p_0\rVert+\tilde{s}\leq(3+1/4z)\varrho ∥ c ∗ − p 0 ∥ + s ~ ≤ ( 3 + 1/4 z ) ϱ Z ≤ ( 4 + 1 / 4 z ) z O P T Q ≤ 1.07 ⋅ 4 z O P T Q Z\leq(4+1/4z)^z\mathrm{OPT}_Q\leq1.07\cdot4^z\mathrm{OPT}_Q Z ≤ ( 4 + 1/4 z ) z OPT Q ≤ 1.07 ⋅ 4 z OPT Q s ~ / ( z N ′ ) \tilde{s}/(zN') s ~ / ( z N ′ ) D D D ∥ c t − c ∗ ∥ ≤ 3 s ~ \lVert c_t-c^*\rVert\leq3\tilde{s} ∥ c t − c ∗ ∥ ≤ 3 s ~ z s ~ z − 1 z\tilde{s}^{z-1} z s ~ z − 1 ( 1 + 1 / ( z N ′ ) ) z − 1 ≤ 1 + 2 / N ′ (1+1/(zN'))^{z-1}\leq1+2/N' ( 1 + 1/ ( z N ′ ) ) z − 1 ≤ 1 + 2/ N ′ ( 1 + 2 / N ′ + 1 / N ′ ) z s ~ z − 1 (1+2/N'+1/\sqrt{N'})z\tilde{s}^{z-1} ( 1 + 2/ N ′ + 1/ N ′ ) z s ~ z − 1 ∥ c t + 1 − c ∗ ∥ 2 \lVert c_{t+1}-c^*\rVert^2 ∥ c t + 1 − c ∗ ∥ 2 2 τ ⋅ 3 s ~ ⋅ z s ~ z − 1 / N ′ = 6 s ~ 2 / N ′ 2\tau\cdot3\tilde{s}\cdot z\tilde{s}^{z-1}/\sqrt{N'}=6\tilde{s}^2/N' 2 τ ⋅ 3 s ~ ⋅ z s ~ z − 1 / N ′ = 6 s ~ 2 / N ′ 4 s ~ 2 / ( z N ′ ) + ( s ~ / ( z N ′ ) ) 2 4\tilde{s}^2/(zN')+(\tilde{s}/(zN'))^2 4 s ~ 2 / ( z N ′ ) + ( s ~ / ( z N ′ ) ) 2 s ~ 2 / N ′ \tilde{s}^2/N' s ~ 2 / N ′ ( 1 + 2 / N ′ + 1 / N ′ ) 2 (1+2/N'+1/\sqrt{N'})^2 ( 1 + 2/ N ′ + 1/ N ′ ) 2 N ′ N' N ′ 2 τ N ′ 2\tau N' 2 τ N ′
E [ ℓ ˉ ( c ˉ ) ] − ℓ ˉ ( c ∗ ) ≤ ( 1 2 + 3 + 1 2 ( 1 + 2 N ′ + 1 N ′ ) 2 + 2 z + 1 2 z 2 N ′ ) z s ~ z N ′ ≤ 8 z s ~ z N ′ . \mathbb{E}[\bar{\ell}(\bar{c})]-\bar{\ell}(c^*)\leq\left(\frac{1}{2}+3+\frac{1}{2}\left(1+\frac{2}{N'}+\frac{1}{\sqrt{N'}}\right)^2+\frac{2}{z}+\frac{1}{2z^2N'}\right)\frac{z\tilde{s}^z}{\sqrt{N'}}\leq\frac{8z\tilde{s}^z}{\sqrt{N'}}. E [ ℓ ˉ ( c ˉ )] − ℓ ˉ ( c ∗ ) ≤ ( 2 1 + 3 + 2 1 ( 1 + N ′ 2 + N ′ 1 ) 2 + z 2 + 2 z 2 N ′ 1 ) N ′ z s ~ z ≤ N ′ 8 z s ~ z . so E [ Ψ Q ( c ˉ ) ] − O P T Q ≤ 1.07 ⋅ 8 z 4 z O P T Q / N ′ ≤ z 4 z O P T Q / N ≤ ϑ O P T Q \mathbb{E}[\Psi_Q(\bar{c})]-\mathrm{OPT}_Q\leq1.07\cdot8z4^z\mathrm{OPT}_Q/\sqrt{N'}\leq z4^z\mathrm{OPT}_Q/\sqrt{N}\leq\vartheta\mathrm{OPT}_Q E [ Ψ Q ( c ˉ )] − OPT Q ≤ 1.07 ⋅ 8 z 4 z OPT Q / N ′ ≤ z 4 z OPT Q / N ≤ ϑ OPT Q Γ \Gamma Γ log Δ ( P ∪ Γ ) \log\Delta(P\cup\Gamma) log Δ ( P ∪ Γ )
Other steps. In the proof of Theorem 1.1 , the number r 0 r_0 r 0 § 11.2 is run in the working metric d ˉ \bar{d} d ˉ ( 1 + η w / z ) z ≤ e η w (1+\eta_w/z)^z\leq e^{\eta_w} ( 1 + η w / z ) z ≤ e η w η / ( 8 k ) \eta/(8k) η / ( 8 k ) 1 + η / ( 8 k ) 1+\eta/(8k) 1 + η / ( 8 k ) j j j ( 1 + η / ( 8 k ) ) j (1+\eta/(8k))^j ( 1 + η / ( 8 k ) ) j ( ( y 1 / z + 2 ) z − y ) / y = ( 1 + 2 y − 1 / z ) z − 1 ((y^{1/z}+2)^z-y)/y=(1+2y^{-1/z})^z-1 (( y 1/ z + 2 ) z − y ) / y = ( 1 + 2 y − 1/ z ) z − 1 y y y 2 k 2k 2 k ( 1 + η / ( 8 k ) ) 2 k ≤ e η / 4 (1+\eta/(8k))^{2k}\leq e^{\eta/4} ( 1 + η / ( 8 k ) ) 2 k ≤ e η /4 H = O ( log Δ ( P ∪ F ) ) H=O(\log\Delta(P\cup F)) H = O ( log Δ ( P ∪ F ))
Accounting. Compared with the exact analysis, the network stretch for pairs that are not Λ \Lambda Λ e η e^\eta e η e 2 η e^{2\eta} e 2 η A A A 1 + O ( η ) 1+O(\eta) 1 + O ( η ) 1 + η c / z 1+\eta_c/z 1 + η c / z η O P T ≤ η O P T \eta\mathrm{OPT}\leq\eta\mathrm{OPT} η OPT ≤ η OPT Theorem 1.1 , the representatives are chosen by costs computed with d ˉ \bar{d} d ˉ η c / z \eta_c/z η c / z η c ≤ ϑ / ( 18 z ) \eta_c\leq\vartheta/(18z) η c ≤ ϑ / ( 18 z ) ( ( 1 + 3 η c ) ( 1 + η c / z ) z ≤ 1 + ϑ / 3 ((1+3\eta_c)(1+\eta_c/z)^z\leq1+\vartheta/3 (( 1 + 3 η c ) ( 1 + η c / z ) z ≤ 1 + ϑ /3 (16) and Ψ ˉ P i ( a i ) ≤ ( 1 + η c / z ) 2 z ( 1 + ϑ / 3 ) 2 min c Ψ ˉ P i ( c ) ≤ ( 1 + ϑ ) min c Ψ ˉ P i ( c ) \bar{\Psi}_{P_i}(a_i)\leq(1+\eta_c/z)^{2z}(1+\vartheta/3)^2\min_c\bar{\Psi}_{P_i}(c)\leq(1+\vartheta)\min_c\bar{\Psi}_{P_i}(c) Ψ ˉ P i ( a i ) ≤ ( 1 + η c / z ) 2 z ( 1 + ϑ /3 ) 2 min c Ψ ˉ P i ( c ) ≤ ( 1 + ϑ ) min c Ψ ˉ P i ( c ) Ψ ˉ P i ( c ) \bar{\Psi}_{P_i}(c) Ψ ˉ P i ( c ) 1 + O ( η ) 1+O(\eta) 1 + O ( η ) O ( η ) ⋅ O P T O(\eta)\cdot\mathrm{OPT} O ( η ) ⋅ OPT η \eta η δ \delta δ O ( δ ) ⋅ O P T O(\delta)\cdot\mathrm{OPT} O ( δ ) ⋅ OPT δ / C \delta/C δ / C δ \delta δ C C C Theorems 1.1 to 1.4 .
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