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 $\epsilon > 0$, a $(6+\epsilon)$-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.

翻译：本研究探讨在两个经典模型中最大化纳什社会福利（即代理人效用的几何平均数）的问题。第一个模型涉及单边偏好，即将一组不可分割物品分配给一组代理人（常见于公平分配研究）。第二个模型处理双边偏好，即一组工人与公司相互匹配，双方对另一方均具有数值化估值（常见于偏好匹配文献）。我们在容量约束下研究这些模型，该约束限制了代理人（或公司）可获得的物品（或工人）数量。针对广泛类别的估值函数，我们为这两个问题开发了常数因子近似算法。具体而言，我们的主要成果如下：(a) 对于任意 $\epsilon > 0$，当代理人具有次模估值时，为单边问题提供 $(6+\epsilon)$-近似算法；(b) 当公司具有次可加估值时，为双边问题提供 $1.33$-近似算法。前者为具有次模估值与容量约束的单边纳什社会福利问题提供了首个常数因子近似算法，而后者改进了现有针对可加估值的 $\sqrt{OPT}$-近似算法。我们在双边场景下的结果还确立了纳什福利与功利主义福利目标之间的计算复杂性分离。此外，我们通过近似难度结果对算法进行了补充分析。