Nash Social Welfare for 2-value Instances

TL;DR

This paper presents a polynomial-time algorithm for maximizing Nash social welfare in allocating indivisible goods among agents with 2-value additive valuations, improving approximation for general cases.

Contribution

The authors develop a polynomial-time algorithm for integrally 2-valued cases and extend it to achieve better approximations for general 2-value instances.

Findings

Polynomial-time algorithm for integrally 2-valued valuations

Improved approximation for general 2-value instances

Enhanced understanding of Nash social welfare maximization

Abstract

This paper is merged with arXiv:2107.08965v2. We refer the reader to the full and updated version. We study the problem of allocating a set of indivisible goods among agents with 2-value additive valuations. Our goal is to find an allocation with maximum Nash social welfare, i.e., the geometric mean of the valuations of the agents. We give a polynomial-time algorithm to find a Nash social welfare maximizing allocation when the valuation functions are integrally 2-valued, i.e., each agent has a value either $1$ or $p$ for each good, for some positive integer $p$. We then extend our algorithm to find a better approximation factor for general 2-value instances.