Bounded Unpopularity Matchings

We investigate the following problem: given a set of jobs and a set of people with preferences over the jobs, what is the optimal way of matching people to jobs? Here we consider the notion of popularity. A matching M is popular if there is no matching M ′ such that more people prefer M ′ to M than the other way around. Determining whether a given instance admits a popular matching and, if so, finding one, was studied in [2]. If there is no popular matching, a reasonable substitute is a matching whose unpopularity is bounded. We consider two mea-sures of unpopularity- unpopularity factor denoted by u(M) and unpopularity margin denoted by g(M). McCutchen recently showed that computing a match-ing M with the minimum value of u(M) or g(M) is NP-hard, and that if G does not admit a popular matching, then we have u(M) ≥ 2 for all matchings M in G. Here we show that a matching M that achieves u(M) = 2 can be computed in O(m n) time (where m is the number of edges in G and n is the number of nodes) provided a certain graph H admits a matching that matches all people. We also describe a sequence of graphs: H = H2,H3,...,Hk such that if Hk admits a matching that matches all people, then we can compute in O(km n) time a matching M such that u(M) ≤ k − 1 and g(M) ≤ n(1 − 2k). Simulation results suggest that our algorithm finds a matching with low unpopularity