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Popular Edges and Dominant Matchings

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Integer Programming and Combinatorial Optimization (IPCO 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9682))

Abstract

Given a bipartite graph \(G = (A \cup B,E)\) with strict preference lists and given an edge \(e^* \in E\), we ask if there exists a popular matching in G that contains \(e^*\). We call this the popular edge problem. A matching M is popular if there is no matching \(M'\) such that the vertices that prefer \(M'\) to M outnumber those that prefer M to \(M'\). It is known that every stable matching is popular; however G may have no stable matching with the edge \(e^*\). In this paper we identify another natural subclass of popular matchings called “dominant matchings” and show that if there is a popular matching that contains the edge \(e^*\), then there is either a stable matching that contains \(e^*\) or a dominant matching that contains \(e^*\). This allows us to design a linear time algorithm for the popular edge problem. When preference lists are complete, we show an \(O(n^3)\) algorithm to find a popular matching containing a given set of edges or report that none exists, where \(n = |A| + |B|\).

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Acknowledgment

Thanks to Chien-Chung Huang for useful discussions which led to the definition of dominant matchings.

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Authors and Affiliations

  1. Reykjavík University, Reykjavik, Iceland

    Ágnes Cseh

  2. Tata Institute of Fundamental Research, Mumbai, India

    Telikepalli Kavitha

Authors
  1. Ágnes Cseh
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  2. Telikepalli Kavitha
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Corresponding author

Correspondence to Ágnes Cseh .

Editor information

Editors and Affiliations

  1. Université de Liège, Liège, Belgium

    Quentin Louveaux

  2. Technische Universität Berlin, Berlin, Germany

    Martin Skutella

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© 2016 Springer International Publishing Switzerland

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Cseh, Á., Kavitha, T. (2016). Popular Edges and Dominant Matchings. In: Louveaux, Q., Skutella, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 2016. Lecture Notes in Computer Science(), vol 9682. Springer, Cham. https://doi.org/10.1007/978-3-319-33461-5_12

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  • DOI: https://doi.org/10.1007/978-3-319-33461-5_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-33460-8

  • Online ISBN: 978-3-319-33461-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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