AI 中文总结
本文针对字典序估值下的纳什福利最大化问题，提出了更优的近似算法与精确多项式框架，同时证明该问题仍存在显著计算难度，并非易事。
AI 中文摘要
在不可分商品上最大化纳什福利是资源分配领域的核心问题。对于加法估值，最知名的近似因子约为 $e^{-1/e}\approx0.692$，且该问题是APX难问题。本文研究字典序估值下的纳什福利最大化问题，其中每个商品的价值均高于所有排名更低商品的总价值。这种巨大差距结构使得偏好几乎是序数的，可能会让人认为该问题会变得简单。然而，我们的研究表明情况更为复杂：尽管字典序估值能带来更强的算法保证，但仍存在显著的计算难度。我们的第一个主要结果是针对字典序估值下加权纳什福利的 $(1/\sqrt{2}-\epsilon)\approx(0.707-\epsilon)$ 近似算法，改进了继承自加法估值的约 $e^{-1/e}$ 的保证。该算法对纳什福利的配置线性规划（configuration LP）进行取整，我们证明其匹配的整性间隙（integrality gap）为 $\sqrt{2}$。我们的第二个主要贡献是针对有序字典序实例和倍增字典序实例的精确多项式时间算法框架。我们引入了一种基于支配关系的分支剪枝方法，证明了兄弟子树之间的互斥性质，并采用基于矩阵的叶子计数论证，当智能体数量为常数时，将剪枝后的递归树约束为多项式规模。最后，我们证明巨大差距并未消除难度：即使对于有序字典序估值，纳什福利最大化仍是NP难问题；即使对于倍增字典序估值，要获得0.9996的近似比也是NP难问题。因此，字典序估值使纳什福利最大化变得更简单，但并非易事：它们在重要情况下能提供更紧的近似和精确算法，但仍需要复杂的技术，且保留了一般加法设置的部分难度。
英文摘要
Maximizing Nash welfare over indivisible goods is a central problem in resource allocation. For additive valuations, the best-known approximation factor is roughly $e^{-1/e}\approx0.692$, and the problem is APX-hard. We study Nash welfare maximization under lexicographic valuations, where every good is worth more than the total value of all lower-ranked goods. This large-gap structure makes preferences almost ordinal, which might suggest that the problem becomes easy. We show, however, that the picture is more nuanced: although lexicographic valuations enable stronger algorithmic guarantees, they retain significant computational hardness. Our first main result is a $(1/\sqrt{2}-ε)\approx(0.707-ε)$-approximation algorithm for weighted Nash welfare under lexicographic valuations, improving over the inherited guarantee of roughly $e^{-1/e}$. The algorithm rounds the configuration LP for Nash welfare, for which we show a matching integrality gap of $\sqrt{2}$. Our second main contribution is an exact algorithmic polynomial time framework for ordered lexicographic instances and doubling lexicographic instances. We introduce a domination-based branch-and-prune method for which we prove a mutual-exclusion property between sibling subtrees and use a matrix-based leaf-counting argument to bound the pruned recursion tree by a polynomial when the number of agents is constant. Finally, we show that large gaps do not eliminate hardness, as Nash welfare maximization is NP-hard even for ordered lexicographic valuations, and it is NP-hard to obtain a $0.9996$-approximation even for doubling lexicographic valuations. Thus, lexicographic valuations make Nash welfare maximization easier, but not easy: they admit tighter approximation and exact algorithms in important cases, yet still require intricate techniques and preserve some of the hardness of the general additive setting.
CommentsAbstract shortened to match arXiv requirements
