Approximating One-Sided and Two-Sided Nash Social Welfare With Capacities
Salil Gokhale, Harshul Sagar, Rohit Vaish, Vignesh Viswanathan, Jatin Yadav
TL;DR
This work investigates maximizing Nash social welfare (NSW) under capacity constraints in two core models: one-sided NSW with indivisible items and two-sided NSW with workers and firms. It delivers constant-factor approximation algorithms—$6+\varepsilon$ for capacitated one-sided NSW with submodular valuations and $1.33$ for capacitated two-sided NSW with subadditive valuations—and extends to a weighted NSW setting via a modified configuration LP achieving $e^{1/e+\varepsilon}$ for additive valuations. A PTAS is obtained for a constant number of firms in the two-sided setting, and the paper establishes APX-hardness results showing NSW is computationally easier than utilitarian welfare in this landscape. The results advance fair-division and matching under capacities by providing scalable algorithms, new analytical techniques (endowed valuations, local swaps), and a clear separation between NSW and utilitarian objectives. These contributions have practical impact on resource allocation problems where capacity constraints and complex valuations are prevalent, and they open avenues for further refinements and extensions to broader constraint families.
Abstract
We study the problem of maximizing Nash social welfare, which is the geometric mean of agents' utilities, in two well-known models. The first model involves one-sided preferences, where a set of indivisible items is allocated among a group of agents (commonly studied in fair division). The second model deals with two-sided preferences, where a set of workers and firms, each having numerical valuations for the other side, are matched with each other (commonly studied in matching-under-preferences literature). We study these models under capacity constraints, which restrict the number of items (respectively, workers) that an agent (respectively, a firm) can receive. We develop constant-factor approximation algorithms for both problems under a broad class of valuations. Specifically, our main results are the following: (a) For any $ε> 0$, a $(6+ε)$-approximation algorithm for the one-sided problem when agents have submodular valuations, and (b) a $1.33$-approximation algorithm for the two-sided problem when the firms have subadditive valuations. The former result provides the first constant-factor approximation algorithm for Nash welfare in the one-sided problem with submodular valuations and capacities, while the latter result improves upon an existing $\sqrt{OPT}$-approximation algorithm for additive valuations. Our result for the two-sided setting also establishes a computational separation between the Nash and utilitarian welfare objectives. We also complement our algorithms with hardness-of-approximation results. Additionally, for the case of additive valuations, we modify the configuration LP of Feng and Li [ICALP 2024] to obtain an $(e^{1/e}+ε)-$ approximation algorithm for weighted two-sided Nash social welfare under capacity constraints.