Fair Allocation of Indivisible Chores: Beyond Additive Costs

TL;DR

This paper investigates fair allocation of indivisible chores with complex cost functions, establishing tight bounds for submodular and subadditive costs, and exploring specific cases like bin packing and job scheduling.

Contribution

It proves new impossibility and approximation bounds for MMS fairness beyond additive costs, and shows constant approximations exist for certain subadditive cost scenarios.

Findings

No algorithm can guarantee better than (n, log m / log log m)-approximation for submodular costs.

Existence of (n, log m)-approximate allocations for subadditive costs, tight asymptotically.

Constant approximate allocations exist for bin packing and job scheduling problems.

Abstract

We study the maximin share (MMS) fair allocation of $m$ indivisible chores to $n$ agents who have costs for completing the assigned chores. It is known that exact MMS fairness cannot be guaranteed, and so far the best-known approximation for additive cost functions is $\frac{13}{11}$ by Huang and Segal-Halevi [EC, 2023]; however, beyond additivity, very little is known. In this work, we first prove that no algorithm can ensure better than $\min\{n,\frac{\log m}{\log \log m}\}$-approximation if the cost functions are submodular. This result also shows a sharp contrast with the allocation of goods where constant approximations exist as shown by Barman and Krishnamurthy [TEAC, 2020] and Ghodsi et al. [AIJ, 2022]. We then prove that for subadditive costs, there always exists an allocation that is $\min\{n,\lceil\log m\rceil\}$-approximation, and thus the approximation ratio is asymptotically tight. Besides multiplicative approximation, we also consider the ordinal relaxation, 1-out-of-$d$ MMS, which was recently proposed by Hosseini et al. [JAIR and AAMAS, 2022]. Our impossibility result implies that for any $d\ge 2$, a 1-out-of-$d$ MMS allocation may not exist. Due to these hardness results for general subadditive costs, we turn to studying two specific subadditive costs, namely, bin packing and job scheduling. For both settings, we show that constant approximate allocations exist for both multiplicative and ordinal relaxations of MMS.