EFX Allocations in Hypergraphs with Girth at Least 4
A recent mathematical proof reveals that envy-free-up-to-any-good (EFX) allocations consistently exist in hypergraph fair division scenarios where the hypergraph's girth is a minimum of 4, applicable even to agents with general monotone valuations. This finding builds on the earlier research by Christodoulou et al. (2023), which examined graph-based fair division, treating agents and goods as vertices and edges, respectively, with only endpoints holding non-zero marginal value. The new results, detailed in a paper on arXiv (2608.03171), indicate that an EFX allocation can be achieved in polynomial time for these hypergraphs. Additionally, the proof extends to multi-hypergraphs with a girth of at least 4, given a specific vertex has incident edges with limited multiplicity. This work addresses a significant unresolved question in fair division regarding the existence of EFX allocations for additive valuations. The authors are researchers in theoretical computer science and economics, contributing to the field of fair allocation algorithms.
Key facts
- The paper proves existence of EFX allocations in hypergraphs with girth at least 4.
- The result applies to agents with general monotone valuations.
- The allocation can be constructed in polynomial time.
- The work extends the multi-hypergraph setting introduced by Christodoulou et al. (2023).
- The proof also covers multi-hypergraphs with girth at least 4 on the simple hypergraph.
- The condition requires a single vertex whose incident edges have multiplicity at most a certain bound.
- The paper is available on arXiv under identifier 2608.03171.
- The problem of EFX existence is a major open problem in fair division.
Entities
Institutions
- arXiv