Greedy Algorithms for Maximizing Nash Social Welfare

TL;DR

This paper investigates simple greedy algorithms for maximizing Nash social welfare in fair division problems, achieving near-optimal or exact solutions in special cases with identical or binary valuations, offering a simpler alternative to complex methods.

Contribution

The paper demonstrates that greedy algorithms can effectively approximate or exactly solve Nash social welfare maximization in specific settings, providing a simpler approach compared to existing sophisticated techniques.

Findings

Greedy algorithm achieves 1.061-approximation for identical valuations.

Exact polynomial-time solution for binary valuations.

Extends to concave valuations, offering new algorithms.

Abstract

We study the problem of fairly allocating a set of indivisible goods among agents with additive valuations. The extent of fairness of an allocation is measured by its Nash social welfare, which is the geometric mean of the valuations of the agents for their bundles. While the problem of maximizing Nash social welfare is known to be APX-hard in general, we study the effectiveness of simple, greedy algorithms in solving this problem in two interesting special cases. First, we show that a simple, greedy algorithm provides a 1.061-approximation guarantee when agents have identical valuations, even though the problem of maximizing Nash social welfare remains NP-hard for this setting. Second, we show that when agents have binary valuations over the goods, an exact solution (i.e., a Nash optimal allocation) can be found in polynomial time via a greedy algorithm. Our results in the binary setting extend to provide novel, exact algorithms for optimizing Nash social welfare under concave valuations. Notably, for the above mentioned scenarios, our techniques provide a simple alternative to several of the existing, more sophisticated techniques for this problem such as constructing equilibria of Fisher markets or using real stable polynomials.