In this paper, we study how to fairly allocate m indivisible chores to n (asymmetric) agents. We consider (weighted) proportionality up to any item (PROPX) and show that a (weighted) PROPX allocation always exists and can be computed efficiently. For chores, we argue that PROPX might be a more reliable relaxation for proportionality by the facts that any PROPX allocation ensures 2-approximation of maximin share (MMS) fairness [Budish, 2011] for symmetric agents and of anyprice share (APS) fairness [Babaioff et al, 2021] for asymmetric agents. APS allocations for chores have not been studied before the current work, and our result implies a 2-approximation algorithm. Another by-product result is that an EFX and a weighted EF1 allocation for indivisible chores exist if all agents have the same ordinal preference, which might be of independent interest. We then consider the partial information setting and design algorithms that only use agents' ordinal preferences to compute approximately PROPX allocations. Our algorithm achieves a 2-approximation for both symmetric and asymmetric agents, and the approximation ratio is optimal. Finally, we study the price of fairness (PoF), i.e., the loss in social welfare by enforcing allocations to be (weighted) PROPX. We prove that the tight ratio for PoF is Theta(n) for symmetric agents and unbounded for asymmetric agents.

翻译：在本文中,我们研究如何公平地将不可分割的杂务分配到n(非对称)物剂。我们考虑(加权)比例,将(加权)比例提升到任何物品(PROPX),并表明(加权)PROPX分配始终存在,而且可以有效计算。对于杂务,我们争辩说,PROPX分配可能更可靠地放松比例,因为任何PROPX分配都确保了对称物剂和任何价格份额(APS)公平[Babaioff等人,20211]的对称性(加权)比例。在目前工作之前,还没有研究过对杂务分配的(加权)PROPX分配,而我们的结果是,对于不可分割的杂杂务分配的加权EFX和EF分配,如果所有物剂都具有相同的或有独立利益。然后我们考虑部分信息设置和设计算法,即仅使用代理人的或定价偏好来计算约PROX分配。我们算法的稳妥性物剂的公平性比,最后是用于Siralimalimex代理商和Simalimalimalimaleximpralimpration。我们进行2-Apractimpress的代理商的最佳定价研究。