（多）超图中的EFX分配
EFX Allocation In (Multi)Hypergraphs
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中文总结 AI 辅助
该研究在（多超）图设置下，证明围长至少为4的超图（主体具一般单调估值）及满足特定条件的多超图中，总存在EFX分配，分别可在多项式和伪多项式时间构造。
中文摘要 AI 辅助
我们研究具有异态单调估值的主体之间不可分割商品的公平分配，这里的公平指无妒忌-至多任何一个商品（EFX）的分配。EFX分配是否总是存在，即使对于具有加性估值的主体，也是公平分配领域的一个重大开放问题。Christodoulou等人（2023）引入了（多超）图设置，其中主体和商品分别由图的顶点和边表示，仅边的端点可对其具有非零边际价值。我们证明，对于围长至少为4且主体具有一般单调估值的超图，总是存在EFX分配，且可在多项式时间内构造。我们将方法推广，证明当简单超图的围长至少为4，且存在单个顶点，其关联边的重数至多为该边大小减2时，多超图也总是允许EFX分配；此情况下的构造需要伪多项式时间。
英文摘要
We study fair allocations of indivisible goods among agents with heterogeneous monotone valuations. As fair we consider the allocations that are envy-free-up-to-any-good (EFX). Finding if EFX alloca- tions always exist, even for agents with additive valuations, is a major open problem in Fair Division. Christodoulou et al. (2023) introduced the (multi-hyper)graph setting, where agents and goods are represented by vertices and edges of a graph, respectively, and only the endpoints of an edge may have non-zero marginal value for it. We show that for hypergraphs with girth at least 4 and agents with general monotone valuations there always exists an EFX allocation and can be constructed in polynomial time. We generalize our approach to also show that multi-hypergraphs with girth (on the simple hypergraph) at least 4 always admit an EFX allocation, as long as there exists a single vertex whose incident edges have multiplicity at most the size of that edge minus 2; our construction in this case needs pseudo-polynomial time.
发表机构
- University of West Attica（西阿提卡大学）
- Athens University of Economics and Business（雅典经济与商业大学）
- Archimedes/Athena RC（阿基米德/雅典娜研究中心）
机构由 AI 辅助整理，请以论文原文为准。
