EFX Allocation In (Multi)Hypergraphs

2026-08-04

AI Summary

This work investigates the existence and construction of envy-free up to any good (EFX) allocations among agents with heterogeneous monotone valuations under hypergraph and multi-hypergraph structures. Focusing on settings where only endpoints of hyperedges derive non-zero marginal value from items, the study establishes—for the first time—that EFX allocations always exist for hypergraphs with girth at least four under general monotone valuations, extending beyond the previously studied additive case. The result is further generalized to multi-hypergraphs satisfying specific multiplicity constraints. By integrating tools from combinatorial graph theory and algorithm design, and leveraging girth restrictions together with edge multiplicity analysis, the authors present a polynomial-time algorithm for constructing EFX allocations in hypergraphs with girth ≥ 4, and a quasi-polynomial-time algorithm for the broader class of generalized multi-hypergraphs.

Abstract

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.

Problem

• EFX allocation
• fair division
• hypergraphs
• indivisible goods
• envy-free

Innovation

• EFX allocation
• hypergraphs
• girth constraint
• monotone valuations
• fair division