Popular edges and dominant matchings

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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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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Keywords

  • Popular matching
  • Matching under preferences
  • Dominant matching

Mathematics Subject Classification

  • 05C70
  • 68W40
  • 05C85