发表机构
CNRS, Université Claude Bernard Lyon 1, INSA Lyon, LIRIS; McGill University; Massachusetts Institute of Technology（法国国家科学研究中心、里昂第一大学、里昂高等师范学院; 麦吉尔大学; 麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究图结构资源的公平分配问题，聚焦$\text{EF1}_{\text{outer}}$公平性，证明无限类非可迹图存在$\text{EF1}_{\text{outer}}$分配，推进树的相关猜想，还证明该分配存在性判定为NP完全，加深图论与公平分配的联系。
AI 中文摘要
本文研究资源以图形式结构化且智能体必须获得连通束的公平分配问题。这种连通性要求从根本上改变了问题，使其比经典对应问题更具挑战性。我们聚焦于公平性概念$\text{EF1}_{\text{outer}}$，其中可通过移除最多一个不使束断开的顶点来消除嫉妒，这是土地分配、网络分配等应用的关键约束。我们的首个结果扩展了先前工作，为无限类非可迹图（即不具有哈密顿路径的图）建立了$\text{EF1}_{\text{outer}}$分配的存在性，回答了一个核心开放问题，并推广了Bilò等人针对可迹图的结果。随后，我们在Chen和Zwicker提出的关于树的$\text{EF1}_{\text{outer}}$谱的猜想上取得进展。最后，我们补充了算法层面的见解，证明即使对于二元加性估值，判定$\text{EF1}_{\text{outer}}$分配的存在性也是NP完全的，从而解决了一个开放的复杂度问题。综上，我们的结果加深了图论与公平分配之间的联系，为研究结构化资源环境中的公平性提供了新工具。
英文摘要
In this paper, we study fair division problems in which resources are structured as graphs and agents must receive connected bundles. This connectivity requirement fundamentally alters the problem, making it significantly more challenging than its classical counterpart. We focus on the fairness notion of $\mathrm{EF1}_{\mathrm{outer}}$, where envy can be eliminated by removing at most one vertex whose deletion does not disconnect the bundle -- a critical constraint for applications such as land division and network allocation. Our first result extends prior work by establishing the existence of $\mathrm{EF1}_{\mathrm{outer}}$ allocations for an infinite family of non-traceable graphs (that is, graphs that do not admit a Hamiltonian path), answering a central open question and generalizing Bilò et al.'s result for traceable graphs. We then make progress on a conjecture concerning the $\mathrm{EF1}_{\mathrm{outer}}$ spectrum of trees due to Chen and Zwicker. Finally, we complement our structural results with algorithmic insights, showing that deciding the existence of an $\mathrm{EF1}_{\mathrm{outer}}$ allocation is NP-complete even for binary additive valuations, thereby resolving an open complexity question. Taken together, our results deepen the connection between graph theory and fair division, and offer new tools for studying fairness in structured resource environments.
