Existence of 2-EFX Allocations of Chores

TL;DR

This paper proves the existence of 2-EFX allocations for indivisible chores among agents with additive disutilities, improving previous bounds and providing a versatile framework for fair division.

Contribution

It introduces a general framework for approximate-EFX allocations, achieving 2-EFX existence for all instances and simplifying proofs of existing results.

Findings

Proved 2-EFX existence for all instances with chores and additive disutilities.

Provided a unified framework for approximate-EFX allocations using local swaps.

Simplified proofs for known results like 2-EFX for bivalued instances and three agents.

Abstract

We study the fair division of indivisible chores among agents with additive disutility functions. We investigate the existence of allocations satisfying the popular fairness notion of envy-freeness up to any chore (EFX), and its multiplicative approximations. The existence of $4$-EFX allocations was recently established by Garg, Murhekar, and Qin (2025). We improve this guarantee by proving the existence of $2$-EFX allocations for all instances with additive disutilities. This approximation was previously known only for restricted instances such as bivalued disutilities (Lin, Wu, and Zhou (2025)) or three agents (Afshinmehr, Ansaripour, Danaei, and Mehlhorn (2024)). We obtain our result by providing a general framework for achieving approximate-EFX allocations. The approach begins with a suitable initial allocation and performs a sequence of local swaps between the bundles of envious and envied agents. For our main result, we begin with an initial allocation that satisfies envy-freeness up to one chore (EF1) and Pareto-optimality (PO); the existence of such an allocation was recently established in a major breakthrough by Mahara (2025). We further demonstrate the strength and generality of our framework by giving simple and unified proofs of existing results, namely (i) $2$-EFX for bivalued instances, (ii) 2-EFX for three agents, (iii) EFX when the number of chores is at most twice the number of agents, and (iv) $4$-EFX for all instances. We expect this framework to have broader applications in approximate-EFX due to its simplicity and generality.