Maximizing Nash Social Welfare under Two-Sided Preferences

TL;DR

This paper explores the computational complexity of maximizing Nash social welfare in two-sided preferences, revealing NP-hardness in general and proposing approximation and parameterized algorithms for specific cases.

Contribution

It is the first systematic study of NSW maximization for two-sided preferences, including complexity results and algorithms for restricted domains.

Findings

Maximizing NSW is NP-hard even with limited capacities and valuations.

Developed approximation algorithms for general two-sided matching.

Provided parameterized algorithms for special cases like symmetric valuations.

Abstract

The maximum Nash social welfare (NSW) -- which maximizes the geometric mean of agents' utilities -- is a fundamental solution concept with remarkable fairness and efficiency guarantees. The computational aspects of NSW have been extensively studied for one-sided preferences where a set of agents have preferences over a set of resources. Our work deviates from this trend and studies NSW maximization for two-sided preferences, wherein a set of workers and firms, each having a cardinal valuation function, are matched with each other. We provide a systematic study of the computational complexity of maximizing NSW for many-to-one matchings under two-sided preferences. Our main negative result is that maximizing NSW is NP-hard even in a highly restricted setting where each firm has capacity 2, all valuations are in the range {0,1,2}, and each agent positively values at most three other agents. In search of positive results, we develop approximation algorithms as well as parameterized algorithms in terms of natural parameters such as the number of workers, the number of firms, and the firms' capacities. We also provide algorithms for restricted domains such as symmetric binary valuations and bounded degree instances.