A Note on EFX Inapproximability for Chores

2026-05-20

AI Summary

This study investigates the existence of constant-factor approximations to envy-freeness up to any good (EFX) in the allocation of indivisible chores under complement-free cost functions. Focusing on monotone subadditive and submodular cost settings, the authors introduce a novel approach combining refined counterexample constructions with a weighted covering model and ordinal preference profiles. They establish the first explicit inapproximability lower bounds for EFX allocations: in a three-agent, six- chore instance, any α-EFX allocation must satisfy α ≥ 2^{1/3} ≈ 1.26 under subadditive costs and α ≥ 20/19 under submodular costs. These results significantly narrow the gap between previously known upper and lower bounds, advancing the understanding of fairness guarantees in chore division under realistic cost structures.

Abstract

We study the approximability of EFX allocations for indivisible chores under complement-free cost functions. The non-existence of exact EFX allocations for general monotone functions for chores is known from \cite{CS24}, and a result of \cite{akrami2026} transfers such comparison-based non-existence results to monotone submodular, and hence subadditive, functions. We strengthen this picture by giving explicit constant-factor inapproximability results for submodular and subadditive functions. Our main construction is a three-agent, six-chore instance with monotone subadditive cost functions for which no $α$-EFX allocation exists for any $1\le α<2^{1/3}\approx 1.26$, thus narrowing the gap with the known upper bound of $2$. The construction is obtained by refining the original counterexample of \cite{CS24} and using the approach of \cite{mackenzie2026}. We also give a weighted-coverage realization of the ordinal profile, yielding an instance in which no $α$-EFX allocation exists for any $1\le α<20/19$ under submodular costs. Thus, even within well-studied complement-free classes, EFX for chores admits nontrivial constant lower bounds on approximability.

Problem

• EFX
• chores
• inapproximability
• submodular
• subadditive

Innovation

• EFX
• chores
• submodular
• subadditive
• inapproximability