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 identical ordering 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 3n^2 regarding EFX. We also study the bi-valued instances, in which agents have at most two cost values on the chores, and provide polynomial time algorithms for the computation of EFX allocation when n=3, and (n-1)-approximation of EFX allocation when n>=4.
翻译:在本文中,我们研究如何公平地将一组不可分割的杂活分配给一组N代理,每个代理都有一般的累加成本功能。由于无法保证不存在无嫉妒(EF)分配,我们考虑对任何项目(EFX)的无嫉妒(EFX)分配概念。与商品EFX分配的(接近)丰硕成果相比,在分配杂活方面鲜为人知。在我们工作之前,在分配杂活方面,已知EFX分配始终为两个代理,或具有相同定购成本功能的一般数目的代理。在一般情况下,不存在关于EFX分配的非三重近似结果。在本文中,我们在这方面取得了一些进展,我们表明对于三个代理,我们总是可以在多边时间对EFX分配进行5倍的认可。关于杂活分配,我们的算法总是计算出在EFXXxx的近似比率上达到3n%2,我们还研究了双值实例,即代理在最大两个成本分配时,XIFSL的NRA值, 和NEFAxxxA的计算时,提供了对工作和Nxxxxxx的计算。