On the existence of EFX allocations in multigraphs
Alkmini Sgouritsa, Minas Marios Sotiriou
TL;DR
The paper tackles the long-standing question of whether EFX allocations exist for fair division with indivisible goods under a multigraph valuation model. By combining cut-and-choose based initial partitions, a three-stage construction to reduce envy, and a final allocation phase that parks remaining unallocated bundles, the authors prove existence of EFX allocations under three structural conditions: bipartite multigraphs, a degree bound of $\lceil \frac{n}{4} \rceil-1$ neighbors per agent, and a girth requirement that the shortest cycle with non-parallel edges has length at least $6$, all for general monotone valuations. The approach hinges on forming an initial EFX orientation, systematically reducing envy via UNP/UNPB constructs, and carefully finalizing assignments to preserve EFX while guaranteeing completion. The results extend prior work on EFX in simple graphs and restricted multigraphs, and align with concurrent findings while offering a unified framework for handling bundles and parallel edges. The paper also discusses limitations and directions for future work, including removing graph restrictions and exploring computational efficiency or approximations to EFX.
Abstract
We study the problem of "fairly" dividing indivisible goods to several agents that have valuation set functions over the sets of goods. As fair we consider the allocations that are envy-free up to any good (EFX), i.e., no agent envies any proper subset of the goods given to any other agent. The existence or not of EFX allocations is a major open problem in Fair Division, and there are only positive results for special cases. [George Christodoulou, Amos Fiat, Elias Koutsoupias, Alkmini Sgouritsa 2023] introduced a restriction on the agents' valuations according to a graph structure: the vertices correspond to agents and the edges to goods, and each vertex/agent has zero marginal value (or in other words, they are indifferent) for the edges/goods that are not adjacent to them. The existence of EFX allocations has been shown for simple graphs with general monotone valuations [George Christodoulou, Amos Fiat, Elias Koutsoupias, Alkmini Sgouritsa 2023], and for multigraphs for restricted additive valuations [Alireza Kaviani, Masoud Seddighin, Amir Mohammad Shahrezaei 2024]. In this work, we push the state-of-the-art further, and show that the EFX allocations always exists in multigraphs and general monotone valuations if any of the following three conditions hold: either (a) the multigraph is bipartite, or (b) each agent has at most $\lceil \frac{n}{4} \rceil -1$ neighbors, where $n$ is the total number of agents, or (c) the shortest cycle with non-parallel edges has length at least 6.