Bounded Unpopularity Matchings
 ChienChung Huang,
 Telikepalli Kavitha,
 Dimitrios Michail,
 Meghana Nasre
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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 by Abraham et al. (SIAM J. Comput. 37(4):1030–1045, 2007). 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 NPhard, 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 in random instances.
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 Title
 Bounded Unpopularity Matchings
 Journal

Algorithmica
Volume 61, Issue 3 , pp 738757
 Cover Date
 20111101
 DOI
 10.1007/s0045301094349
 Print ISSN
 01784617
 Online ISSN
 14320541
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Matching with preferences
 Popularity
 Approximation algorithms
 Industry Sectors
 Authors

 ChienChung Huang ^{(1)}
 Telikepalli Kavitha ^{(2)}
 Dimitrios Michail ^{(3)}
 Meghana Nasre ^{(2)}
 Author Affiliations

 1. MaxPlanckInstitut für Informatik, Saarbrücken, Germany
 2. Indian Institute of Science, Bangalore, India
 3. Department of Informatics and Telematics, Harokopion University of Athens, Athens, Greece