Two Problems in Max-Size Popular Matchings

Abstract

We study popular matchings in the many-to-many matching problem, which is given by a graph $$G = (V,E)$$ and, for every agent $$u\in V$$ , a capacity $$\textsf {cap}(u) \ge 1$$ and a preference list strictly ranking her neighbors. A matching in G is popular if it weakly beats every matching in a majority vote when agents cast votes for one matching versus the other according to their preferences. First, we show that when $$G = (A\cup B,E)$$ is bipartite, e.g., when matching students to courses, every pairwise stable matching is popular. In particular, popular matchings are guaranteed to exist. Our main contribution is to show that a max-size popular matching in G can be computed in linear time by the 2-level Gale–Shapley algorithm, which is an extension of the classical Gale–Shapley algorithm. We prove its correctness via linear programming. Second, we consider the problem of computing a max-size popular matching in $$G = (V,E)$$ (not necessarily bipartite) when every agent has capacity 1, e.g., when matching students to dorm rooms. We show that even when G admits a stable matching, this problem is $$\mathsf {NP}$$ -hard, which is in contrast to the tractability result in bipartite graphs.