Maximizing Nash Social Welfare in 2-Value Instances

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 $p\mathrm{/}q=4\mathrm{/}5$.