Popular Edges and Dominant Matchings

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9682)


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 \(n = |A| + |B|\).



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


  1. 1.
    Biró, P., Irving, R.W., Manlove, D.F.: Popular matchings in the marriage and roommates problems. In: Calamoneri, T., Diaz, J. (eds.) CIAC 2010. LNCS, vol. 6078, pp. 97–108. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  2. 2.
    Cseh, Á., Huang, C.-C., Kavitha, T.: Popular matchings with two-sided preferences and one-sided ties. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9134, pp. 367–379. Springer, Heidelberg (2015)Google Scholar
  3. 3.
    Dias, V.M.F., da Fonseca, G.D., de Figueiredo, C.M.H., Szwarcfiter, J.L.: The stable marriage problem with restricted pairs. Theor. Comput. Sci. 306, 391–405 (2003)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Feder, T.: A new fixed point approach for stable networks and stable marriages. J. Comput. Syst. Sci. 45, 233–284 (1992)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Feder, T.: Network flow and 2-satisfiability. Algorithmica 11, 291–319 (1994)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Monthly 69, 9–15 (1962)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gale, D., Sotomayor, M.: Some remarks on the stable matching problem. Discrete Appl. Math. 11, 223–232 (1985)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gärdenfors, P.: Match making: assignments based on bilateral preferences. Behav. Sci. 20, 166–173 (1975)CrossRefGoogle Scholar
  9. 9.
    Gusfield, D., Irving, R.W.: The Stable Marriage Problem: Structure and Algorithms. MIT Press, Cambridge (1989)MATHGoogle Scholar
  10. 10.
    Huang, C.-C., Kavitha, T.: Popular matchings in the stable marriage problem. Inf. Comput. 222, 180–194 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Irving, R.W., Leather, P., Gusfield, D.: An efficient algorithm for the “optimal” stable marriage. J. ACM 34, 532–543 (1987)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kavitha, T.: A size-popularity tradeoff in the stable marriage problem. SIAM J. Comput. 43, 52–71 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Knuth, D.: Mariages Stables. Les Presses de L’Université de Montréal (1976). English translation in Stable Marriage and its Relation to Other Combinatorial Problems. CRM Proceedings and Lecture Notes, vol. 10. American Mathematical Society (1997)Google Scholar
  14. 14.
    McDermid, E., Irving, R.W.: Popular matchings: structure and algorithms. J. Comb. Optim. 22(3), 339–359 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Rothblum, U.G.: Characterization of stable matchings as extreme points of a polytope. Math. Program. 54, 57–67 (1992)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Reykjavík UniversityReykjavikIceland
  2. 2.Tata Institute of Fundamental ResearchMumbaiIndia

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