A Better-than-$e^{1/e}$ Approximation Algorithm for Nash Social Welfare under Additive Valuations

2026-07-14

AI Summary

This work addresses the problem of maximizing Nash social welfare under additive valuations, for which the best-known approximation ratio has long been stuck at $e^{1/e}$. We propose a novel algorithmic framework that integrates combinatorial optimization with refined resource allocation analysis, thereby breaking this longstanding theoretical barrier. Our approach yields an algorithm with an approximation ratio of $e^{1/e} - c$ for some positive constant $c$, marking the first improvement over the previous state-of-the-art bound. This result constitutes a significant advancement in the approximability of Nash social welfare maximization, demonstrating a tangible enhancement in solution quality over prior methods.

Abstract

We present an $(e^{1/e} - c)$-approximation algorithm for maximizing Nash social welfare under additive valuations, for some constant $c > 0$. This result improves upon the previous best-known approximation factor of $e^{1/e}$ [Barman, Krishnamurthy and Vaish, EC 2018].

Problem

• Nash Social Welfare
• Approximation Algorithm
• Additive Valuations
• e^{1/e}

Innovation

• Nash Social Welfare
• approximation algorithm
• additive valuations
• improved approximation ratio