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.

翻译：我们研究 $m$ 件不可分割家务在 $n$ 个代理人之间的最大最小份额（MMS）公平分配问题，其中代理人完成分配的家务会产生成本。已知精确的MMS公平性无法保证，且迄今为止，对于可加性成本函数的最佳近似比是 Huang 和 Segal-Halevi [EC, 2023] 提出的 $\frac{13}{11}$；然而，在超越可加性的情形下，相关研究甚少。本文首先证明，若成本函数为子模函数，则任何算法都无法保证优于 $\min\{n,\frac{\log m}{\log \log m}\}$ 的近似比。该结果也与商品分配形成鲜明对比——Barman 和 Krishnamurthy [TEAC, 2020] 以及 Ghodsi 等人 [AIJ, 2022] 已证明商品分配中存在常数近似比。随后我们证明，对于次可加成本函数，始终存在一个 $\min\{n,\lceil\log m\rceil\}$ 近似的分配方案，因此该近似比是渐近紧的。除乘法近似外，我们还考虑序数松弛，即近期由 Hosseini 等人 [JAIR and AAMAS, 2022] 提出的1-out-of-$d$ MMS。我们的不可行性结果表明，对于任意 $d\ge 2$，1-out-of-$d$ MMS 分配可能不存在。鉴于一般次可加成本函数的这些困难性结果，我们转而研究两种特殊的次可加成本函数：装箱问题和作业调度问题。针对这两种场景，我们均证明了在MMS的乘法和序数松弛下均存在常数近似分配方案。