Popular Matchings in the Stable Marriage Problem

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)


The input is a bipartite graph \(G = (\mathcal{A}\cup\mathcal{B}, E)\) where each vertex \(u \in \mathcal{A}\cup\mathcal{B}\) ranks its neighbors in a strict order of preference. A matching M * is said to be popular if there is no matching M such that more vertices are better off in M than in M *. We consider the problem of computing a maximum cardinality popular matching in G. It is known that popular matchings always exist in such an instance G, however the complexity of computing a maximum cardinality popular matching was not known so far. In this paper we give a simple characterization of popular matchings when preference lists are strict and a sufficient condition for a maximum cardinality popular matching. We then show an O(mn 0) algorithm for computing a maximum cardinality popular matching in G, where m = |E| and \(n_0 = \min(|\mathcal{A}|,|\mathcal{B}|)\).


Bipartite Graph Stable Match Maximum Cardinality Preference List Simple Characterization 
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  1. 1.
    Abraham, D.J., Irving, R.W., Kavitha, T., Mehlhorn, K.: Popular matchings. SIAM Journal on Computing 37(4), 1030–1045 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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
  3. 3.
    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. American Mathematical Monthly 69, 9–15 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gärdenfors, P.: Match making: assignments based on bilateral preferences. Behavioural Sciences 20, 166–173 (1975)CrossRefGoogle Scholar
  5. 5.
    Gusfield, D., Irving, R.W.: The Stable Marriage Problem: Structure and Algorithms. MIT Press, Cambridge (1989)zbMATHGoogle Scholar
  6. 6.
    Kavitha, T., Mestre, J., Nasre, M.: Popular mixed matchings. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 574–584. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Knuth, D.E.: Mariages Stables. Les Presses de L’Université de Montreal (1976)Google Scholar
  8. 8.
    Mahdian, M.: Random popular matchings. In: Proceedings of the 7th ACM Conference on Electronic-Commerce, pp. 238–242 (2006)Google Scholar
  9. 9.
    Manlove, D.F., Sng, C.: Popular matchings in the capacitated house allocation problem. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 492–503. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    McCutchen, M.: The least-unpopularity-factor and least-unpopularity-margin criteria for matching problems with one-sided preferences. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 593–604. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  11. 11.
    Mestre, J.: Weighted popular matchings. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 715–726. Springer, Heidelberg (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Humboldt-Universität zu BerlinGermany
  2. 2.Tata Institute of Fundamental ResearchIndia

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