Maximizing Nash Social Welfare in 2-Value Instances: Delineating Tractability

TL;DR

This paper characterizes when maximizing Nash social welfare in 2-value instances is computationally feasible, providing polynomial algorithms for certain cases and proving NP-hardness for others based on valuation parameters.

Contribution

It offers a complete complexity characterization of Nash social welfare maximization in 2-value instances, including polynomial algorithms and NP-hardness results depending on valuation parameters.

Findings

Polynomial-time algorithm for integral valuations ($q=1$)

Algorithm for half-integral valuations ($q=2$)

NP-hardness for $q \, \geq 3$

Abstract

We study the problem of allocating a set of indivisible goods among a set of agents with \emph{2-value additive valuations}. In this setting, each good is valued either $1$ or $p/q$, for some fixed co-prime numbers $p,q\in \mathbb{N}$ such that $1\leq q < p$. Our goal is to find an allocation maximizing the \emph{Nash social welfare} (\NSW), i.e., the geometric mean of the valuations of the agents. In this work, we give a complete characterization of polynomial-time tractability of \NSW\ maximization that solely depends on the values of $q$. We start by providing a rather simple polynomial-time algorithm to find a maximum \NSW\ allocation when the valuation functions are \emph{integral}, that is, $q=1$. We then exploit more involved techniques to get an algorithm producing a maximum \NSW\ allocation for the \emph{half-integral} case, that is, $q=2$. Finally, we show it is \classNP-hard to compute an allocation with maximum \NSW\ whenever $q\geq3$.