We study the problem of allocating divisible bads (chores) among multiple agents with additive utilities when monetary transfers are not allowed. The competitive rule is known for its remarkable fairness and efficiency properties in the case of goods. This rule was extended to chores in prior work by Bogomolnaia, Moulin, Sandomirskiy, and Yanovskaya (2017). The rule produces Pareto optimal and envy-free allocations for both goods and chores. In the case of goods, the outcome of the competitive rule can be easily computed. Competitive allocations solve the Eisenberg-Gale convex program; hence the outcome is unique and can be approximately found by standard gradient methods. An exact algorithm that runs in polynomial time in the number of agents and goods was given by Orlin (2010). In the case of chores, the competitive rule does not solve any convex optimization problem; instead, competitive allocations correspond to local minima, local maxima, and saddle points of the Nash social welfare on the Pareto frontier of the set of feasible utilities. The Pareto frontier may contain many such points; consequently, the competitive rule's outcome is no longer unique. In this paper, we show that all the outcomes of the competitive rule for chores can be computed in strongly polynomial time if either the number of agents or the number of chores is fixed. The approach is based on a combination of three ideas: all consumption graphs of Pareto optimal allocations can be listed in polynomial time; for a given consumption graph, a candidate for a competitive utility profile can be constructed via an explicit formula; each candidate can be checked for competitiveness, and the allocation can be reconstructed using a maximum flow computation. Our algorithm gives an approximately-fair allocation of indivisible chores by the rounding technique of Barman and Krishnamurthy (2018).

翻译：我们研究了在无货币转移情况下，具有可加性效用的多个代理之间分配可分坏品（家务）的问题。竞争性规则在商品分配中以其卓越的公平性和效率特性而闻名。Bogomolnaia、Moulin、Sandomirskiy 和 Yanovskaya（2017）先前的工作将该规则扩展到家务分派。该规则对商品和家务都能产生帕累托最优且无嫉妒的分配方案。对于商品而言，竞争性规则的结果易于计算：竞争性分配求解的是艾森伯格-盖尔凸规划，因此结果唯一，并且可通过标准梯度方法近似求得。Orlin（2010）提出了在代理和商品数量上具有多项式时间复杂度的精确算法。对于家务分派，竞争性规则不求解任何凸优化问题；相反，竞争性分配对应于可行效用集帕累托前沿上纳什社会福利的局部最小值、局部最大值和鞍点。帕累托前沿可能包含多个此类点，因此竞争性规则的结果不再唯一。本文证明，当代理数量或家务数量固定时，家务竞争性规则的所有结果均可在强多项式时间内计算得出。该方法基于三个思路的结合：所有帕累托最优分配的消费图可在多项式时间内枚举；对于给定消费图，可通过显式公式构建竞争性效用配置的候选解；每个候选解可经竞争性检验，并通过最大流计算重构分配。我们的算法通过 Barman 和 Krishnamurthy（2018）的舍入技术，为不可分家务提供了近似公平的分配方案。