A Note on EFX Inapproximability for Chores

TL;DR

This paper establishes explicit constant-factor inapproximability bounds for EFX allocations in chores with submodular and subadditive costs, showing limitations of existing fair division guarantees.

Contribution

It provides the first explicit inapproximability results for EFX chores in submodular and subadditive settings, narrowing the gap with known upper bounds.

Findings

No $eta$-EFX allocation exists for $eta<2^{1/3}$ in a three-agent, six-chore instance.

No $eta$-EFX allocation exists for $eta<20/19$ in a weighted-coverage instance.

The results demonstrate fundamental limitations in approximating EFX for chores with certain cost functions.

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 $\alpha$-EFX allocation exists for any $1\le \alpha<2^{1/3}\approx 1.26$, 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 $\alpha$-EFX allocation exists for any $1\le \alpha<20/19$ under submodular costs. Thus, even within well-studied complement-free classes, EFX for chores admits nontrivial constant lower bounds on approximability.