A Note on Approximating Weighted Nash Social Welfare with Additive Valuations

TL;DR

This paper presents a simple, effective algorithm achieving a constant-factor approximation for the weighted Nash Social Welfare problem with additive valuations, matching the best unweighted case approximation.

Contribution

It introduces the first $O(1)$-approximation algorithm for weighted Nash Social Welfare with additive valuations, using a natural LP and a randomized rounding technique.

Findings

Achieves an approximation ratio of approximately 1.445+ε.

Uses a configuration LP and a randomized Shmoys-Tardos rounding algorithm.

The approximation ratio matches the best known for unweighted cases.

Abstract

We give the first $O(1)$-approximation for the weighted Nash Social Welfare problem with additive valuations. The approximation ratio we obtain is $e^{1/e} + \epsilon \approx 1.445 + \epsilon$, which matches the best known approximation ratio for the unweighted case. Both our algorithm and analysis are simple. We solve a natural configuration LP for the problem, and obtain the allocation of items to agents using a randomized version of the Shmoys-Tardos rounding algorithm developed for unrelated machine scheduling problems. In the analysis, we show that the approximation ratio of the algorithm is at most the worst gap between the Nash social welfare of the optimum allocation and that of an EF1 allocation, for an unweighted Nash Social Welfare instance with identical additive valuations. This was shown to be at most $e^{1/e} \approx 1.445$ by Barman, Krishnamurthy and Vaish, leading to our approximation ratio.