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 $q\geq3$.

翻译：本研究探讨在具有\emph{2值可加估值}的智能体集合间分配不可分割物品的问题。在此设定下，每个物品的估值固定为$1$或$p/q$，其中$p,q\in \mathbb{N}$为固定互质整数且满足$1\leq q < p$。我们的目标是找到最大化\emph{纳什社会福利}（\NSW）的分配方案，即智能体估值的几何平均值。本文通过完全刻画仅取决于$q$值的多项式时间可解性，给出了\NSW\最大化问题的完整分类。首先，当估值函数为\emph{整值}（即$q=1$）时，我们提出一种较为简单的多项式时间算法来求解最大\NSW\分配。随后，通过运用更复杂的技术，我们设计了针对\emph{半整值}情形（即$q=2$）生成最大\NSW\分配的算法。最后，我们证明当$q\geq3$时，计算最大\NSW\分配是\classNP困难的。