Polynomial-Time Algorithms for Fair Orientations of Chores

TL;DR

This paper presents polynomial-time algorithms for finding fair chore orientations in graphs, resolving a conjecture and highlighting a complexity separation between chores and goods, with implications for fair division problems.

Contribution

It provides the first polynomial-time algorithms for EF1 and EFX chore orientations, proving NP-completeness for multigraphs and graphs with only goods.

Findings

Polynomial-time algorithms for EF1 and EFX chore orientations.

NP-completeness results for multigraphs and goods-only graphs.

A complexity separation between chores and goods in fair orientation problems.

Abstract

This paper addresses the problem of finding fair 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 have EFX orientations, and conjectured that deciding whether graphs containing only chores have EFX orientations is NP-complete. We resolve this conjecture by giving polynomial-time algorithms that find EF1 and EFX orientations of graphs containing only chores if they exist, even if there are self-loops. Remarkably, our result demonstrates a surprising separation between the case of goods and the case of chores, because deciding whether graphs containing only goods have EFX orientations was shown to be NP-complete by Christodoulou et al. (EC, 2023). In addition, we show the EF1 and EFX orientation problems for multigraphs to be NP-complete.