竞争-单射分配：支持列表结构与最大锚点EFX₀证书
Rival-Injective Allocations: Support-List Structure and Maximum-Anchor EFX$_0$ Certificates
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中文总结 AI 辅助
该研究针对非负加性估值下的全物品EFX₀分配，提出竞争-单射分配的支持列表结构，通过反向贪心着色构造最大锚点EFX₀分配，给出见证族并研究其识别问题，未解决四智能体十物品全物品EFX₀的存在性。
中文摘要 AI 辅助
我们研究非负加性估值下的完整分配，聚焦于无嫉妒至多任何物品（EFX₀）的全物品形式，通过物品的正支持集展开分析。我们定义了竞争-单射（RI）端点所有权：每个具有非空正支持集的物品被分配给对其估值为正的智能体，且每个有序观察者-所有者对最多被一个物品使用。RI所有权恰好是对那些在至少两个智能体中具有重叠正支持集的物品构成的图进行的真列表着色。对于具有任意例外物品的双支持物品，固定例外所有者会产生一个充分必要的双容量准则，以及一个精确的有限域所有者约束满足问题（CSP）。每个满足每个智能体自身束的价值至少为其最高估值单物品的RI分配，均为全物品EFX₀分配。通过指定的最高单物品锚点的单射选择，结合剩余支持列表退化，可通过反向贪心着色在O(nm²)时间内构造此类分配。我们给出两个显式见证族：固定4×10支持面上的非空相对开、22维锥，以及对每个n≥4、具有两个通用支持物品和m=(n-1)(n-2)+2个物品的族。这两者均不满足无锚点列表退化，且与此处对比的显式纯多重图或已发表的高围长/受控重数假设无关。我们还研究了最大锚点证书类的识别问题，其总体复杂度仍未解决。精确的列表着色和双容量结果涉及RI所有权，而非一般EFX₀的存在性；无约束的四智能体、十物品全物品EFX₀问题仍未解决。
英文摘要
We study complete allocations under nonnegative additive valuations through the positive supports of goods, focusing on the all-good form of envy-freeness up to any good (EFX$_0$). We define rival-injective (RI) endpoint ownership: every good with nonempty positive support is assigned to an agent who values it positively, and every ordered observer--owner pair is used by at most one good. RI ownership is exactly proper list coloring of the graph joining goods whose positive supports overlap in at least two agents. For pair-supported goods with arbitrary exceptional goods, fixing the exceptional owners yields a necessary-and-sufficient pair-capacity criterion and an exact finite-domain owner constraint satisfaction problem (CSP). Every RI allocation in which each agent's own bundle is worth at least her maximum-valued singleton is all-good EFX$_0$. A specified injective choice of maximum-singleton anchors, together with residual support-list degeneracy, constructs such an allocation by reverse greedy coloring in $O(nm^2)$ time. We give two explicit witness families: a nonempty relatively open, 22-dimensional cone on a fixed $4\times10$ support face, and a family for every $n\ge4$ with two universal-support goods and $m=(n-1)(n-2)+2$ goods. Both fail unanchored list degeneracy and are not implied by the explicit pure-multigraph or published high-girth/controlled-multiplicity hypotheses compared here. We also study recognition of the maximum-anchor certificate class, leaving its general complexity unresolved. The exact list-coloring and pair-capacity results concern RI ownership, not general EFX$_0$ existence; unrestricted four-agent, ten-good all-good EFX$_0$ remains unresolved.
发表机构
- University of Electronic Science and Technology of China（电子科技大学）
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