发表机构
Siebel School of Computing and Data Science, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校塞贝尔计算与数据科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文给出强 EFX 可定向图的多项式时间刻画：连通图是强 EFX 可定向的当且仅当其为二部图，或其块分解中恰有一个非二部块且满足特定度条件，解决了开放问题。
AI 中文摘要
离散公平分配问题是将一组离散商品在多个智能体之间公平分配的问题。在此背景下，最受追捧的公平性概念之一是“无嫉妒至任意商品”（EFX）。2023 年，Christodoulou、Fiat、Koutsoupias 和 Sgouritsa 引入了图形估价（graphical valuation）的概念，其中公平分配问题用一个简单图表示，图的顶点代表智能体，边代表商品，每个顶点仅对其关联的边赋予价值。他们证明了 EFX 分配总是存在，而确定 EFX 定向的存在性则是 NP 难的。他们提出了一个开放问题：确定哪些图无论估价如何都总是允许 EFX 定向。这些图被称为强 EFX 可定向图，由 Zeng 和 Mehta 于 2025 年首次研究，他们证明了所有此类图的色数至多为 3，且二部图无论估价如何总是允许 EFX 定向。在本手稿中，我们通过给出强 EFX 可定向图的多项式时间刻画，最终解决了该问题。特别地，我们证明了一个连通图 $G$ 是强 EFX 可定向的当且仅当以下两种情况之一成立：（1）$G$ 是二部图，或（2）$G$ 的块分解中恰好包含一个非二部块 $B$，并且存在一个顶点 $v \in B$，使得 $v$ 在 $B$ 内的度为 2，且 $G-v$ 是二部图。该证明由人工智能发现，人类干预以将问题分解为适当的子问题。
英文摘要
Discrete fair division is the problem of dividing a discrete set of goods among agents in a fair manner. In this setting, one of the most sought-after notions of fairness is envy-freeness up to any good (EFX). In 2023, Christodoulou, Fiat, Koutsoupias, and Sgouritsa introduced the idea of a graphical valuation, where the fair division problem is represented by a simple graph where vertices are the agents and the edges are the goods, and each vertex only values incident edges. They showed that an EFX allocation always exists, while determining the existence of an EFX orientation is NP-hard. They posed a open question of determining which graphs always admit an EFX orientation regardless of valuation. These graphs, called strongly EFX orientable graphs, were first studied by Zeng and Mehta in 2025, who demonstrated that all such graphs have chromatic number at most 3, and bipartite graphs always admit an EFX orientation regardless of valuation. In this manuscript, we finish resolving this question by giving a polynomial time characterization of strongly EFX orientable graphs. In particular, we show that a connected graph $G$ is strongly EFX orientable if and only if either of the following is true: (1) $G$ is bipartite, or (2) the block decomposition of $G$ contains exactly one nonbipartite block $B$, and there exists a vertex $v \in B$ such that the degree of $v$ within $B$ is 2 and $G-v$ is bipartite. This proof was discovered by AI, with human intervention to break the problem into the appropriate subproblems.
Comments16 pages, 9 figures
