Almost EFX in Hypergraphs

2026-06-25

AI Summary

This work addresses the fair allocation of indivisible goods under a hypergraph structure, where hyperedges represent items and vertices represent agents, with non-zero marginal value for an item only assigned to its incident agents. For heterogeneous monotone valuation functions, the study combines combinatorial constructions with greedy strategies to achieve various approximate envy-free up to any good (EFX) allocations in polynomial or pseudo-polynomial time. Key contributions include the first existence proof of 2/3-EFX allocations for hypergraphs with edge multiplicity two, along with a simpler construction achieving √2/2-EFX; for hypergraphs of girth at least three, it establishes EF2X under general monotone valuations and √2/2-EFX under subadditive valuations; and for edge multiplicity two, it further obtains EF3X and 2/3-EFX under additive valuations.

Abstract

We study the existence of envy-free-up-to-any-good (EFX) allocations of indivisible goods among agents with heterogeneous monotone valuations. 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. Our work simplifies and extends previous results of Kaviani et al. (Alireza Kaviani, Masoud Seddighin, Amir Mohammad Shahrezaei. Almost Envy-Free Allocation of Indivisible Goods: A Tale of Two Valuations. WINE 2024) in this domain. First, we provide a simpler construction of EF2X allocation for general monotone valuations in hypergraphs with girth at least 3. We extend our ideas when the multiplicity of each edge is 2 and show that an EF3X allocation always exists for additive valuations. Both results can be constructed in polynomial time. Regarding EFX approximations, we provide a simpler construction for $\frac{\sqrt{2}}{2}$-EFX allocations in hypergraphs of girth at least 3 under subadditive valuations. We push the state-of-the-art by establishing the existence of $\frac{2}{3}$-EFX allocations for additive valuations when the edge multiplicity is 2. Both of the latter results can be constructed in pseudo-polynomial time. By addressing these multi-hypergraph settings, our work contributes to the ongoing effort to resolve the existence of EFX in increasingly general and applicable domains.

Problem

• EFX
• hypergraphs
• envy-free allocation
• indivisible goods
• monotone valuations

Innovation

• EFX allocation
• hypergraphs
• approximate fairness
• monotone valuations
• polynomial-time construction