Weijiang Hu | Searcharxiv

arXiv2026

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.

cs.DS↗

arXiv2026

New Convex Programming Technique for Nash Social Welfare and Scheduling

We propose a new convex programming relaxation for the weighted Nash social welfare (NSW) problem that achieves a matching $(e^{1/e}\approx 1.445)$-approximation via the rounding algorithm of Feng and Li. Unlike the exponential-size configuration LP used in prior work, our formulation can be converted into a compact linear program of polynomial size, incurring only an additive loss of $\ln(1+ε)$ in the objective. This allows the program to be solved directly using standard LP solvers, without the ellipsoid method or dual separation oracles. In the unweighted case, we show that our convex program is equivalent to the restricted-spending Fisher market convex program of Cole and Gkatzelis, yielding a constructive proof that its integrality gap is exactly $e^{1/e}$. With a minor modification, our analysis also gives a simple proof of the $e^{1/e}$ EF1 gap for the identical agent setting. Finally, we show that our convex programming technique extends to two unrelated machine scheduling problems, recovering the best-known approximation ratios with simpler analyses.

cs.DS↗