In this paper we study how to fairly allocate a set of m indivisible chores to a group of n agents, each of which has a general additive cost function on the items. Since envy-free (EF) allocation is not guaranteed to exist, we consider the notion of envy-freeness up to any item (EFX). In contrast to the fruitful results regarding the (approximation of) EFX allocations for goods, very little is known for the allocation of chores. Prior to our work, for the allocation of chores, it is known that EFX allocations always exist for two agents, or general number of agents with IDO cost functions. For general instances, no non-trivial approximation result regarding EFX allocation is known. In this paper we make some progress in this direction by showing that for three agents we can always compute a 5-approximation of EFX allocation in polynomial time. For n>=4 agents, our algorithm always computes an allocation that achieves an approximation ratio of O(n^2) regarding EFX.

中文翻译：

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不可分割的杂务的近似 EFX 分配

在本文中，我们研究如何将一组 m 个不可分割的杂务公平地分配给一组 n 个代理，每个代理在项目上都有一个通用的附加成本函数。由于不保证无嫉妒 (EF) 分配存在，我们将无嫉妒的概念考虑到任何项目 (EFX)。与商品 EFX 分配（近似）的丰硕成果相反，对杂务分配知之甚少。在我们的工作之前，对于琐事的分配，众所周知，EFX 分配始终存在于两个代理或具有 IDO 成本函数的代理的一般数量。对于一般情况，关于 EFX 分配的非平凡近似结果是未知的。在本文中，我们在这个方向上取得了一些进展，表明对于三个代理，我们总是可以在多项式时间内计算 EFX 分配的 5 近似值。