Skip to main content

Popular edges and dominant matchings

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.


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

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10


  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)

    MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  4. Feder, T.: A new fixed point approach for stable networks and stable marriages. J. Comput. Syst. Sci. 45, 233–284 (1992)

    Article  MathSciNet  Google Scholar 

  5. Feder, T.: Network flow and 2-satisfiability. Algorithmica 11, 291–319 (1994)

    Article  MathSciNet  Google Scholar 

  6. Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Mon. 69, 9–15 (1962)

    Article  MathSciNet  Google Scholar 

  7. Gale, D., Sotomayor, M.: Some remarks on the stable matching problem. Discrete Appl. Math. 11, 223–232 (1985)

    Article  MathSciNet  Google Scholar 

  8. Gärdenfors, P.: Match making: assignments based on bilateral preferences. Behav. Sci. 20, 166–173 (1975)

    Article  Google Scholar 

  9. Gusfield, D., Irving, R.W.: The Stable Marriage Problem: Structure and Algorithms. MIT Press, Cambridge (1989)

    MATH  Google Scholar 

  10. Huang, C.-C., Kavitha, T.: Popular matchings in the stable marriage problem. Inf. Comput. 222, 180–194 (2013)

    Article  MathSciNet  Google Scholar 

  11. Irving, R.W., Leather, P., Gusfield, D.: An efficient algorithm for the “optimal” stable marriage. J. ACM 34, 532–543 (1987)

    Article  MathSciNet  Google Scholar 

  12. Kavitha, T.: A size-popularity tradeoff in the stable marriage problem. SIAM J. Comput. 43, 52–71 (2014)

    Article  MathSciNet  Google Scholar 

  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)

  14. McDermid, E., Irving, R.W.: Popular matchings: structure and algorithms. J. Comb. Optim. 22(3), 339–359 (2011)

    Article  MathSciNet  Google Scholar 

  15. Rothblum, U.G.: Characterization of stable matchings as extreme points of a polytope. Math. Program. 54, 57–67 (1992)

    Article  MathSciNet  Google Scholar 

Download references


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

Author information

Authors and Affiliations


Corresponding author

Correspondence to Ágnes Cseh.

Additional information

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cseh, Á., Kavitha, T. Popular edges and dominant matchings. Math. Program. 172, 209–229 (2018).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Popular matching
  • Matching under preferences
  • Dominant matching

Mathematics Subject Classification

  • 05C70
  • 68W40
  • 05C85