APX-Hardness of Maximizing Nash Social Welfare with Indivisible Items

TL;DR

This paper proves that maximizing Nash social welfare with indivisible items is APX-hard, establishing the computational difficulty of finding approximate solutions despite existing constant factor approximations.

Contribution

It demonstrates the APX-hardness of the problem, complementing prior results that showed constant factor approximability.

Findings

The problem is APX-hard, indicating no polynomial-time approximation scheme exists unless P=NP.

Existing algorithms can achieve constant factor approximations, but optimal solutions are computationally hard.

The result clarifies the complexity landscape of fair division with indivisible items.

Abstract

We study the problem of allocating a set of indivisible items to agents with additive utilities to maximize the Nash social welfare. Cole and Gkatzelis recently proved that this problem admits a constant factor approximation. We complement their result by showing that this problem is APX-hard.