We study the problem of fairly and efficiently allocating indivisible chores among agents with additive disutility functions. We consider the widely-used envy-based fairness properties of EF1 and EFX, in conjunction with the efficiency property of fractional Pareto-optimality (fPO). Existence (and computation) of an allocation that is simultaneously EF1/EFX and fPO are challenging open problems, and we make progress on both of them. We show existence of an allocation that is - EF1+fPO, when there are three agents, - EF1+fPO, when there are at most two disutility functions, - EFX+fPO, for three agents with bivalued disutilities. These results are constructive, based on strongly polynomial-time algorithms. We also investigate non-existence and show that an allocation that is EFX+fPO need not exist, even for two agents.

翻译：我们研究了在具有可加性负效用函数的智能体之间公平且高效地分配不可分割家务劳动的问题。我们考虑了广泛使用的基于嫉妒的公平性属性EF1和EFX，并结合了分数帕累托最优性（fPO）的效率属性。同时满足EF1/EFX和fPO的分配的存在性（及计算）是极具挑战性的开放问题，我们在两个方面均取得了进展。我们证明了以下分配的存在性：针对三个智能体的EF1+fPO分配；针对最多两种负效用函数的EF1+fPO分配；针对具有双值负效用的三个智能体的EFX+fPO分配。这些结果具有构造性，基于强多项式时间算法。我们还探讨了不存在性，并表明即使对于两个智能体，EFX+fPO分配也可能不存在。