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Bounded Unpopularity Matchings

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Abstract

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 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 in random instances.

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Authors and Affiliations

  1. Max-Planck-Institut für Informatik, Saarbrücken, Germany

    Chien-Chung Huang

  2. Indian Institute of Science, Bangalore, India

    Telikepalli Kavitha & Meghana Nasre

  3. Department of Informatics and Telematics, Harokopion University of Athens, Athens, Greece

    Dimitrios Michail

Authors
  1. Chien-Chung Huang
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  2. Telikepalli Kavitha
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  3. Dimitrios Michail
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  4. Meghana Nasre
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Corresponding author

Correspondence to Chien-Chung Huang.

Additional information

A preliminary version of this work appeared in 11th Scandinavian Workshop on Algorithm Theory (SWAT 08).

Part of this work was done when C.-C. Huang was visiting Max-Planck-Institut für Informatik, Saarbrücken, Germany.

Work of T. Kavitha was done as part of the DST-MPG partner group “Efficient Graph Algorithms” at IISc Bangalore.

Part of this work by D. Michail was carried out during the tenure of an ERCIM “Alain Bensoussan” Fellowship Programme.

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Huang, CC., Kavitha, T., Michail, D. et al. Bounded Unpopularity Matchings. Algorithmica 61, 738–757 (2011). https://doi.org/10.1007/s00453-010-9434-9

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