Estimating the Nash Social Welfare for coverage and other submodular valuations

TL;DR

This paper develops approximation algorithms for maximizing Nash Social Welfare in allocation problems with submodular valuations, extending known results to coverage and matroid-based functions.

Contribution

It introduces a novel approximation approach for the Nash Social Welfare problem with specific classes of submodular valuations, previously unresolved.

Findings

Achieves a (1-rac{1}{e})^2-approximation for coverage valuations.

Provides approximation guarantees for sums of matroid rank functions.

Extends results to certain matching-based valuation classes.

Abstract

We study the Nash Social Welfare problem: Given $n$ agents with valuation functions $v_i:2^{[m]} \rightarrow {\mathbb R}$, partition $[m]$ into $S_1,\ldots,S_n$ so as to maximize $(\prod_{i=1}^{n} v_i(S_i))^{1/n}$. The problem has been shown to admit a constant-factor approximation for additive, budget-additive, and piecewise linear concave separable valuations; the case of submodular valuations is open. We provide a $\frac{1}{e} (1-\frac{1}{e})^2$-approximation of the {\em optimal value} for several classes of submodular valuations: coverage, sums of matroid rank functions, and certain matching-based valuations.