A Polynomial-Time Algorithm for EFX Orientations of Chores

2025-01-23

AI Summary

This paper studies the EFX orientation problem on chore graphs: assigning a direction to each edge (chore) such that agents derive zero marginal utility from non-adjacent chores and the resulting allocation satisfies EFX fairness. Contrary to the prior conjecture of NP-completeness, we present the first exact O(n + m)-time algorithm, proving that EFX orientability on chore graphs is in P. Our work reveals a fundamental dichotomy in EFX decidability between goods and chores. We further initiate the systematic study of graphs with self-loops and multiple edges: EFX orientability is polynomial-time solvable with self-loops, yet becomes NP-complete with multiple edges. Methodologically, we combine graph-theoretic modeling, a greedy orientation strategy, and structured inductive proofs—achieving both theoretical rigor and computational efficiency.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

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.

Problem

• Fair Division
• Household Chores
• EFX Allocation

Innovation

• EFX Orientation
• Household Chores Allocation
• Non-NP Complete