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arXiv · 2608.30203

Rival-Injective Allocations: Support-List Structure and Maximum-Anchor EFX$_0$ Certificates

Junshuo Wang
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Abstract

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.

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Junshuo Wang. 2026-08-31. Rival-Injective Allocations: Support-List Structure and Maximum-Anchor EFX$_0$ Certificates. https://arxiv.org/abs/2608.30203

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