Bounded Unpopularity Matchings

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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 measures of unpopularity - unpopularity factor denoted by u(M) and unpopularity margin denoted by g(M). McCutchen recently showed that computing a matching 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 \(O(m\sqrt{n})\) 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 = H 2, H 3,...,H k such that if H k admits a matching that matches all people, then we can compute in \(O(km\sqrt{n})\) time a matching M such that u(M) ≤ k − 1 and \(g(M) \le n(1-\frac{2}{k})\). Simulation results suggest that our algorithm finds a matching with low unpopularity.