On Approximate Envy-Freeness for Indivisible Chores and Mixed Resources

TL;DR

This paper explores fair division of chores and mixed resources, proving NP-completeness of envy-free allocation, providing a polynomial-time EF1 algorithm, and establishing existence of envy-free allocations in mixed resource settings.

Contribution

It introduces a polynomial-time algorithm for EF1 allocation of chores, proves NP-completeness of envy-free chores division, and shows existence of envy-free allocations in mixed resources.

Findings

Envy-free chores division is NP-complete.

A polynomial-time EF1 algorithm for chores is provided.

Envy-free allocations exist for mixed indivisible and divisible resources.

Abstract

We study the fair allocation of undesirable indivisible items, or chores. While the case of desirable indivisible items (or goods) is extensively studied, with many results known for different notions of fairness, less is known about the fair division of chores. We study the envy-free division of chores, and make three contributions. First, we show that determining the existence of an envy-free allocation is NP-complete, even in the simple case when agents have binary additive valuations. Second, we provide a polynomial-time algorithm for computing an allocation that satisfies envy-freeness up to one chore (EF1), correcting an existing proof in the literature. A straightforward modification of our algorithm can be used to compute an EF1 allocation for doubly monotone instances (wherein each agent can partition the set of items into objective goods and objective chores). Our third result applies to a mixed resources model consisting of indivisible items and a divisible, undesirable heterogeneous resource (i.e., a bad cake). We show that there always exists an allocation that satisfies envy-freeness for mixed resources (EFM) in this setting, complementing a recent result of Bei et al. (Art. Int. 2021) for indivisible goods and divisible cake.