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) allocations are 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 identical ordering cost functions. For general instances, no non-trivial approximation result regarding EFX allocation is known. In this paper, we make progress in this direction by providing several polynomial time algorithms for the computation of EFX and approximately EFX allocations. We show that for three agents we can always compute a 4.45-approximation of EFX allocation. For n>=4 agents, our algorithm always computes a (3n^2-n)-approximation. We also study the bi-valued instances, in which agents have at most two cost values on the chores. For three agents, we provide an algorithm for the computation of EFX allocations. For n>=4 agents, we present algorithms for the computation of partial EFX allocations with at most n-1 unallocated items; and (n-1)-approximation of EFX allocations.

翻译：本文研究如何将一组m个不可分割的杂务公平分配给n个智能体，每个智能体对物品具有一般可加性成本函数。由于无嫉妒（EF）分配无法保证存在，我们考虑任意物品无嫉妒（EFX）这一概念。与物品的EFX分配（近似）研究取得丰硕成果相反，关于杂务分配的结果鲜有报道。在本文工作之前，已知杂务分配中EFX分配仅对两个智能体或具有相同序成本函数的任意数量智能体存在。对于一般实例，尚无关于EFX分配的非平凡近似结果。本文通过提出多个用于计算EFX及近似EFX分配的多项式时间算法，在该方向上取得进展。我们证明：对于三个智能体，总能计算出EFX分配的4.45近似解；对于n≥4个智能体，算法总能得到(3n²-n)近似解。我们还研究了二值实例（智能体对杂务最多有两种成本值）。针对三个智能体，我们给出计算EFX分配的算法；对于n≥4个智能体，我们提出可计算至多包含n-1个未分配物品的部分EFX分配算法，以及EFX分配的(n-1)近似算法。