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

We study the problem of allocating a set of indivisible goods among a set of agents with 2-value additive valuations. In this setting, each good is valued either $1$ or $\frac{p}{q}$, for some fixed co-prime numbers $p,q\in N$ such that $1\leq q < p$, and the value of a bundle is the sum of the values of the contained goods. Our goal is to find an allocation which maximizes the 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 integral, that is, $q=1$. We then exploit more involved techniques to get an algorithm producing a maximum NSW allocation for the half-integral case, that is, $q=2$. Finally, we show that such an improvement cannot be further extended to the case $q=3$; indeed, we prove that it is NP-hard to compute an allocation with maximum NSW whenever $q\geq 3$.