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


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


Popular matching Matching under preferences Dominant matching 

Mathematics Subject Classification

05C70 68W40 05C85 



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: Proceedings of 7th CIAC, pp. 97–108 (2010)zbMATHGoogle Scholar
  2. 2.
    Cseh, Á., Huang, C.-C., Kavitha, T.: Popular matchings with two-sided preferences and one-sided ties. In: Proceedings of 42nd ICALP, pp. 367–379 (2015)CrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Feder, T.: A new fixed point approach for stable networks and stable marriages. J. Comput. Syst. Sci. 45, 233–284 (1992)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Feder, T.: Network flow and 2-satisfiability. Algorithmica 11, 291–319 (1994)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Mon. 69, 9–15 (1962)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gale, D., Sotomayor, M.: Some remarks on the stable matching problem. Discrete Appl. Math. 11, 223–232 (1985)MathSciNetCrossRefGoogle 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)zbMATHGoogle Scholar
  10. 10.
    Huang, C.-C., Kavitha, T.: Popular matchings in the stable marriage problem. Inf. Comput. 222, 180–194 (2013)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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, volume 10 of CRM Proceedings and Lecture Notes, 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)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Rothblum, U.G.: Characterization of stable matchings as extreme points of a polytope. Math. Program. 54, 57–67 (1992)MathSciNetCrossRefGoogle Scholar

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

Personalised recommendations