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 $(2+\sqrt{6})$-approximation of EFX allocation. For $n\ge 4$ 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 $n\ge 4$ 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.

在本文中，我们研究如何将一组m个不可分割的家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加性成本函数。由于不保证存在无羡慕（EF）分配，因此我们考虑任何项目（EFX）的无羡慕概念。与商品 EFX 分配（近似）方面的丰硕成果相比，人们对家务分配知之甚少。在我们的工作之前，对于杂务的分配，我们知道 EFX 分配总是存在于两个代理或具有相同排序成本函数的一般数量的代理中。对于一般情况，没有关于 EFX 分配的重要近似结果是已知的。在本文中，我们通过提供几种用于计算 EFX 和近似 EFX 分配的多项式时间算法，在这个方向上取得了进展。我们证明，对于三个代理，我们总是可以计算出$（2+\sqrt{6}）$-EFX 分配的近似值。为了$n\ge 4$代理，我们的算法总是计算$（3n^2-n）$-近似。我们还研究了双值实例，其中代理最多有两个家务成本值。对于三个代理，我们提供了一种计算 EFX 分配的算法。为了$n\ge 4$代理，我们提出了计算部分 EFX 分配的算法，最多$（n-1）$未分配的项目；和$（n-1）$-EFX 分配的近似值。