EFX Allocations on Some Multi-graph Classes

TL;DR

This paper extends the existence of EFX allocations from simple graphs to multi-graphs, providing polynomial-time algorithms for certain classes with cancellable valuations, advancing fair division theory.

Contribution

It proves the existence and computability of EFX allocations in multi-graphs for specific classes, generalizing previous results from simple graphs.

Findings

EFX allocations exist in bipartite multi-graphs

EFX allocations exist in multi-trees with monotone valuations

EFX allocations exist in multi-graphs with certain girth conditions

Abstract

The existence of EFX allocations is one of the most significant open questions in fair division. Recent work by Christodolou, Fiat, Koutsoupias, and Sgouritsa ("Fair allocation in graphs", EC 2023) establishes the existence of EFX allocations for graphical valuations, when agents are vertices in a graph, items are edges, and each item has zero value for all agents other than those at its end-points. Thus in this setting, each good has non-zero value for at most two agents, and there is at most one good valued by any pair of agents. This marks one of the few cases when an exact and complete EFX allocation is known to exist for arbitrary agents. In this work, we extend these results to multi-graphs, when each pair of vertices can have more than one edge between them. The existence of EFX allocations in multi-graphs is a natural open question given their existence in simple graphs. We show that EFX allocations exist, and can be computed in polynomial time, for agents with cancellable valuations in the following cases: (i) bipartite multi-graphs, (ii) multi-trees with monotone valuations, and (iii) multi-graphs with girth $(2t-1)$, where $t$ is the chromatic number of the multi-graph. The existence in multi-cycles follows from (i) and (iii).