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 $q\geq 3$.

翻译：我们研究在具有2-值加性估值的智能体集合中分配一组不可分割物品的问题。在该设定中，每个物品的价值为$1$或$\frac{p}{q}$，其中$p,q\in N$为固定互质数且满足$1\leq q < p$，而一个捆绑包的价值等于所含物品价值之和。我们的目标是找到最大化纳什社会福利（NSW）（即智能体估值的几何均值）的分配方案。本文给出了完全依赖于$q$值、关于NSW最大化多项式时间可处理性的完整刻画。我们首先针对估值函数为整数（即$q=1$）的情形，提出一个相当简单的多项式时间算法来寻找最大NSW分配。随后利用更复杂的技术，针对半整数情形（即$q=2$）设计出能产生最大NSW分配的算法。最后证明该改进无法进一步推广至$q=3$的情形：事实上，我们证明了当$q\geq 3$时，计算最大NSW分配是NP难的。