Santa Claus and Weighted Nash Social Welfare with Subadditive Valuations

We study two fairness objectives for allocating $m$ indivisible items among $n$ agents with subadditive valuations: the max-min objective, also known as the Santa Claus problem, which maximizes the smallest value of an agent, and the weighted Nash social welfare, which maximizes the weighted geometric mean of the agents' values. For the max-min allocation problem, we give a randomized polynomial-time $O(\sqrt n)$-approximation algorithm for subadditive valuations with demand queries, improving the $2n-1$ bound of Khot and Ponnuswami. For additive valuations and for submodular valuations with value queries, we give simpler $O(\sqrt n)$-approximation algorithms. For additive valuations, this improves the $O(\sqrt{n\log n})$ bound of Saha and Srinivasan. For these two classes, $O(m^\varepsilon)$-approximation algorithms are also known for every constant $\varepsilon>0$. Our ratios depend only on $n$ and match, up to constants, the $Ω(\sqrt n)$ integrality gap of the configuration LP. For the weighted Nash social welfare problem with subadditive valuations and demand queries, we give an $O(Γ(w))$-approximation algorithm, where $Γ(w)=\exp\bigl((\ln n-H(w))/2\bigr)$ and $H(w)$ is the entropy of the weight vector $w$. The factor $Γ(w)$ equals $1$ for equal weights and is at most $\sqrt n$ for all weights. We show that this ratio is tight for algorithms analyzed against the configuration LP: for every weight vector, the configuration LP has an integrality gap of $Θ(Γ(w))$, even for XOS valuations, and this gap is $Ω(\sqrt n)$ when one agent has almost all the weight. For submodular valuations, the configuration LP has a constant integrality gap for all weights.