Maximizing Nash Social Welfare in 2-Value Instances

TL;DR

This paper studies the problem of allocating indivisible goods to agents with 2-value additive valuations to maximize Nash social welfare, providing polynomial algorithms for special cases and hardness results for general cases.

Contribution

It introduces a polynomial-time algorithm for optimal allocation when p divides q and establishes NP-hardness and APX-hardness results for other cases.

Findings

Optimal allocation computed in polynomial time when p divides q.

NP-hardness when p and q are coprime with p ≥ 3.

Approximation ratio of at most 1.0345 for general cases.

Abstract

We consider the problem of maximizing the Nash social welfare when allocating a set $\mathcal{G}$ of indivisible goods to a set $\mathcal{N}$ of agents. We study instances, in which all agents have 2-value additive valuations: The value of every agent $i \in \mathcal{N}$ for every good $j \in \mathcal{G}$ is $v_{ij} \in \{p,q\}$, for $p,q \in \mathbb{N}$, $p \le q$. Maybe surprisingly, we design an algorithm to compute an optimal allocation in polynomial time if $p$ divides $q$, i.e., when $p=1$ and $q \in \mathbb{N}$ after appropriate scaling. The problem is \classNP-hard whenever $p$ and $q$ are coprime and $p \ge 3$. In terms of approximation, we present positive and negative results for general $p$ and $q$. We show that our algorithm obtains an approximation ratio of at most 1.0345. Moreover, we prove that the problem is \classAPX-hard, with a lower bound of $1.000015$ achieved at $p/q = 4/5$.