Bounded Unpopularity Matchings

• Authors: Chien-Chung Huang;Telikepalli Kavitha;Dimitrios Michail;Meghana Nasre
• Venue: SWAT '08 Proceedings of the 11th Scandinavian workshop on Algorithm Theory
• Year: 2008

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 Mis popular if there is no matching M茂戮驴 such that more people prefer M茂戮驴 to Mthan 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 unpopularityis bounded. We consider two measures of unpopularity - unpopularity factordenoted by u(M) and unpopularity margindenoted by g(M). McCutchen recently showed that computing a matching Mwith the minimum value of u(M) or g(M) is NP-hard, and that if Gdoes not admit a popular matching, then we have u(M) 茂戮驴 2 for all matchings Min G.Here we show that a matching Mthat achieves u(M) = 2 can be computed in $O(m\sqrt{n})$ time (where mis the number of edges in Gand nis the number of nodes) provided a certain graph Hadmits a matching that matches all people. We also describe a sequence of graphs: H= H2, H3,...,Hksuch that if Hkadmits a matching that matches all people, then we can compute in $O(km\sqrt{n})$ time a matching Msuch that u(M) ≤ k茂戮驴 1 and $g(M) \le n(1-\frac{2}{k})$. Simulation results suggest that our algorithm finds a matching with low unpopularity.