arXiv2023
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.