We investigate the existence of fair and efficient allocations of indivisible chores to asymmetric agents who have unequal entitlements or weights. We consider the fairness notion of weighted envy-freeness up to one chore (wEF1) and the efficiency notion of Pareto-optimality (PO). The existence of EF1 and PO allocations of chores to symmetric agents is a major open problem in discrete fair division, and positive results are known only for certain structured instances. In this paper, we study this problem for a more general setting of asymmetric agents and show that an allocation that is wEF1 and PO exists and can be computed in polynomial time for instances with: - Three types of agents, where agents with the same type have identical preferences but can have different weights. - Two types of chores, where the chores can be partitioned into two sets, each containing copies of the same chore. For symmetric agents, our results establish that EF1 and PO allocations exist for three types of agents and also generalize known results for three agents, two types of agents, and two types of chores. Our algorithms use a weighted picking sequence algorithm as a subroutine; we expect this idea and our analysis to be of independent interest.

翻译：我们研究在具有不平等权益或权重的非对称主体间分配不可分割杂务时，公平且高效的分配方案的存在性问题。我们考虑加权无嫉妒性至多一个杂务（wEF1）的公平概念与帕累托最优性（PO）的效率概念。对对称主体而言，杂务的EF1与PO分配存在性是离散公平分配领域的一个重大开放问题，目前仅在特定结构实例中有正面结论。本文针对更一般的非对称主体设置研究该问题，证明对于以下情况，存在可在多项式时间内计算的wEF1且PO的分配方案：- 三种类型的主体，其中相同类型的主体具有相同偏好但权重可能不同。- 两种类型的杂务，即杂务可划分为两个集合，每个集合包含相同杂务的副本。对于对称主体，我们的结果确立了三种类型主体情形下EF1与PO分配的存在性，同时推广了关于三个主体、两种类型主体及两种类型杂务的已知结论。我们的算法采用加权选取序列算法作为子程序；预计该思路及分析将具有独立研究价值。