Approximating One-Sided and Two-Sided Nash Social Welfare With Capacities

2024-11-21

arXiv.org

AI Summary

This paper studies the capacity-constrained Nash social welfare (NSW) maximization problem under two canonical models: (1) one-sided allocation of indivisible items, where agents have submodular utilities; and (2) two-sided matching between workers and firms, where both sides hold subadditive valuations. We present the first $(6+varepsilon)$-approximation algorithm for the one-sided model and a $1.33$-approximation for the two-sided model—both constitute the state-of-the-art and break the prior $sqrt{mathrm{OPT}}$ barrier. Our techniques integrate alternating-path analysis, local search, and bi-scale pricing, specifically tailored to handle nonlinear utility structures. Furthermore, we establish tight inapproximability lower bounds, revealing a fundamental computational distinction between NSW and utilitarian welfare.

Technology Category

• Multiagent Systems: Mechanism Design
• Search and Optimization: Non-convex Optimization
• Game Theory and Economic Paradigms: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Computational social science

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 $epsilon>0$, 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.

Problem

• Maximizing Nash social welfare with capacity constraints
• Approximating fair division with submodular agent valuations
• Improving matching algorithms for two-sided subadditive valuations

Innovation

• Constant-factor approximation for submodular valuations
• 1.33-approximation algorithm for two-sided matching
• Handles capacity constraints in Nash welfare optimization