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Mathematical Programming

, Volume 172, Issue 1–2, pp 209–229 | Cite as

Popular edges and dominant matchings

  • Ágnes CsehEmail author
  • Telikepalli Kavitha
Full Length Paper Series B
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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 \(n = |A| + |B|\).

Keywords

Popular matching Matching under preferences Dominant matching 

Mathematics Subject Classification

05C70 68W40 05C85 

Notes

Acknowledgements

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

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Copyright information

© Springer-Verlag GmbH Germany and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.Hungarian Academy of Sciences and Corvinus University of BudapestBudapestHungary
  2. 2.Tata Institute of Fundamental ResearchMumbaiIndia

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