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A Polynomial-Time Algorithm for EFX Orientations of Chores

This paper addresses the problem of finding EFX orientations of graphs of chores, in which each vertex corresponds to an agent, each edge corresponds to a chore, and a chore has zero marginal utility to an agent if its corresponding edge is not incident to the vertex corresponding to the agent. Recently, Zhou~et~al.~(IJCAI,~2024) analyzed the complexity of deciding whether graphs containing a mixture of goods and chores admit EFX orientations, and conjectured that deciding whether graphs containing only chores admit EFX orientations is NP-complete. In this paper, we resolve this conjecture by exhibiting a polynomial-time algorithm that finds an EFX orientation of a graph containing only chores if one exists, even if the graph contains self-loops. Remarkably, our first result demonstrates a surprising separation between the case of goods and the case of chores, because deciding whether graphs containing only goods admit EFX orientations of goods was shown to be NP-complete by Christodoulou et al.~(EC,~2023). In addition, we show the analogous decision problem for multigraphs to be NP-complete.

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