Approximating Nash Social Welfare under Rado Valuations

TL;DR

This paper develops the first constant-factor approximation algorithms for maximizing Nash social welfare under Rado valuations, covering both symmetric and asymmetric cases with bounded weight ratios.

Contribution

It introduces the first constant-factor approximation algorithms for symmetric and asymmetric Rado valuation cases in Nash social welfare maximization.

Findings

First constant-factor approximation for symmetric Rado valuations.

First constant-factor approximation for asymmetric Rado valuations with bounded weight ratio.

Progress towards general valuation classes in fair division problems.

Abstract

We consider the problem of approximating maximum Nash social welfare (NSW) while allocating a set of indivisible items to $n$ agents. The NSW is a popular objective that provides a balanced tradeoff between the often conflicting requirements of fairness and efficiency, defined as the weighted geometric mean of agents' valuations. For the symmetric additive case of the problem, where agents have the same weight with additive valuations, the first constant-factor approximation algorithm was obtained in 2015. This led to a flurry of work obtaining constant-factor approximation algorithms for the symmetric case under mild generalizations of additive, and $O(n)$-approximation algorithms for more general valuations and for the asymmetric case. In this paper, we make significant progress towards both symmetric and asymmetric NSW problems. We present the first constant-factor approximation algorithm for the symmetric case under Rado valuations. Rado valuations form a general class of valuation functions that arise from maximum cost independent matching problems, including as special cases assignment (OXS) valuations and weighted matroid rank functions. Furthermore, our approach also gives the first constant-factor approximation algorithm for the asymmetric case under Rado valuations, provided that the maximum ratio between the weights is bounded by a constant.