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Popular edges and dominant matchings

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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 identifying the set of popular edges. 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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Acknowledgements

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

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Authors and Affiliations

  1. Hungarian Academy of Sciences and Corvinus University of Budapest, Budapest, Hungary

    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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Correspondence to Ágnes Cseh.

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Á. Cseh was supported by the Hungarian Academy of Sciences under its Momentum Programme (LP2016-3/2016), its János Bolyai Research Scholarship and OTKA Grant K108383. Part of this work was carried out while she was visiting TIFR.

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Cseh, Á., Kavitha, T. Popular edges and dominant matchings. Math. Program. 172, 209–229 (2018). https://doi.org/10.1007/s10107-017-1183-y

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  • DOI: https://doi.org/10.1007/s10107-017-1183-y

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