Popular Matchings in the Capacitated House Allocation Problem

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Abstract

We consider the problem of finding a popular matching in the Capacitated House Allocation problem (CHA). An instance of CHA involves a set of agents and a set of houses. Each agent has a preference list in which a subset of houses are ranked in strict order, and each house may be matched to a number of agents that must not exceed its capacity. A matching M is popular if there is no other matching M′ such that the number of agents who prefer their allocation in M′ to that in M exceeds the number of agents who prefer their allocation in M to that in M′. Here, we give an \(O(\sqrt{C}n_1+m)\) algorithm to determine if an instance of CHA admits a popular matching, and if so, to find a largest such matching, where C is the total capacity of the houses, n 1 is the number of agents and m is the total length of the agents’ preference lists. For the case where preference lists may contain ties, we give an \(O((\sqrt{C}+n_1)m)\) algorithm for the analogous problem.