Popular Matchings with Two-Sided Preferences and One-Sided Ties

  • Ágnes CsehEmail author
  • Chien-Chung HuangEmail author
  • Telikepalli KavithaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9134)


We are given a bipartite graph \(G = (A \cup B, E)\) where each vertex has a preference list ranking its neighbors: in particular, every \(a \in A\) ranks its neighbors in a strict order of preference, whereas the preference lists of \(b \in B\) may contain ties. A matching M is popular if there is no matching \(M'\) such that the number of vertices that prefer \(M'\) to M exceeds the number that prefer M to \(M'\). We show that the problem of deciding whether G admits a popular matching or not is \(\mathsf {NP}\)-hard. This is the case even when every \(b \in B\) either has a strict preference list or puts all its neighbors into a single tie. In contrast, we show that the problem becomes polynomially solvable in the case when each \(b \in B\) puts all its neighbors into a single tie. That is, all neighbors of b are tied in b’s list and and b desires to be matched to any of them. Our main result is an \(O(n^2)\) algorithm (where \(n = |A \cup B|\)) for the popular matching problem in this model. Note that this model is quite different from the model where vertices in B have no preferences and do not care whether they are matched or not.


Bipartite Graph Maximum Match Prefer Post Stable Match Edge Label 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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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)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Berman, P., Karpinski, M., Scott, A.D.: Approximation hardness of short symmetric instances of MAX-3SAT. Electronic Colloquium on Computational Complexity Report, number 49 (2003)Google Scholar
  3. 3.
    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
  4. 4.
    Dulmage, A., Mendelsohn, N.: Coverings of bipartite graphs. Canadian Journal of Mathematics 10, 517–534 (1958)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Gärdenfors, P.: Match making: assignments based on bilateral preferences. Behavioural Science 20, 166–173 (1975)CrossRefGoogle Scholar
  6. 6.
    Graham, R.L., Grötschel, M., Lovasz, L., (eds.) The Handbook of Combinatorics, chapter 3, Matchings and Extensions, by W. R. Pulleyblank, pp. 179–232. North Holland (1995)Google Scholar
  7. 7.
    Huang, C.-C., Kavitha, T.: Popular Matchings in the Stable Marriage Problem. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 666–677. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  8. 8.
    Kavitha, T.: Popularity vs Maximum cardinality in the stable marriage setting. In: Proceedings of the 23rd SODA, pp. 123–134 (2012)Google Scholar
  9. 9.
    Kavitha, T., Mestre, J., Nasre, M.: Popular Mixed Matchings. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 574–584. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  10. 10.
    Kavitha, T., Nasre, M.: Note: Optimal popular matchings. Discrete Applied Mathematics 157(14), 3181–3186 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Mahdian, M.: Random popular matchings. In: Proceedings of the 7th EC, pp. 238–242 (2006)Google Scholar
  12. 12.
    Manlove, D.F., Sng, C.T.S.: 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
  13. 13.
    McCutchen, R.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
  14. 14.
    McDermid, E., Irving, R.W.: Popular Matchings: Structure and Algorithms. In: Ngo, H.Q. (ed.) COCOON 2009. LNCS, vol. 5609, pp. 506–515. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  15. 15.
    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 2015

Authors and Affiliations

  1. 1.TU BerlinBerlinGermany
  2. 2.Chalmers UniversityGöteborgSweden
  3. 3.Tata Institute of Fundamental ResearchMumbaiIndia

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