Approximating Nash Social Welfare by Matching and Local Search

TL;DR

This paper presents new approximation algorithms for the Nash social welfare problem with submodular valuations, achieving near-optimal guarantees and combining fairness with efficiency.

Contribution

It introduces a simple deterministic $(4+ ext{epsilon})$-approximation algorithm for NSW under submodular valuations and extends results to asymmetric cases and fairness constraints.

Findings

Achieves a $(4+ ext{epsilon})$-approximation for NSW with submodular valuations.

Provides an $e ( ext{omega} + 2 + ext{epsilon})$-approximation for the asymmetric variant.

Shows that $1/2$-EFX envy-freeness can be combined with a constant-factor NSW approximation.

Abstract

For any $\varepsilon>0$, we give a simple, deterministic $(4+\varepsilon)$-approximation algorithm for the Nash social welfare (NSW) problem under submodular valuations. We also consider the asymmetric variant of the problem, where the objective is to maximize the weighted geometric mean of agents' valuations, and give an $e (\omega + 2 + \varepsilon)$-approximation if the ratio between the largest weight and the average weight is at most $\omega$. We also show that the $1/2$-EFX envy-freeness property can be attained simultaneously with a constant-factor approximation. More precisely, we can find an allocation in polynomial time that is both $1/2$-EFX and a $(8+\varepsilon)$-approximation to the symmetric NSW problem under submodular valuations.