Algorithms for Competitive Division of Chores

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 case goods. This was extended to chores by Bogomolnaia, Moulin, Sandomirskiy, Yanovskaya. For both goods chores, produces Pareto optimal envy-free allocations. In goods, outcome can be easily computed. Competitive allocations solve Eisenberg-Gale convex program; hence unique approximately found standard gradient methods. An exact algorithm that runs polynomial time number given Orlin. does any optimization problem; instead, correspond local minima, maxima, saddle points Nash social welfare on frontier set feasible utilities. may contain many such and, consequently, no longer unique. this paper, we show all outcomes computed strongly if either or fixed. approach based a combination three ideas: consumption graphs listed time; graph, candidate utility profile constructed via an explicit formula; each checked competitiveness allocation reconstructed using maximum flow computation. Our immediately gives approximately-fair indivisible rounding technique Barman Krishnamurthy. Funding: work supported National Science Foundation (CNS 1518941); Lady Davis Fellowship Trust, Hebrew University Jerusalem; H2020 European Research Council (740435); Linde Institute at Caltech.