pp 1–27 | Cite as

Two Problems in Max-Size Popular Matchings

  • Florian Brandl
  • Telikepalli KavithaEmail author


We study popular matchings in the many-to-many matching problem, which is given by a graph \(G = (V,E)\) and, for every agent \(u\in V\), a capacity \(\textsf {cap}(u) \ge 1\) and a preference list strictly ranking her neighbors. A matching in G is popular if it weakly beats every matching in a majority vote when agents cast votes for one matching versus the other according to their preferences. First, we show that when \(G = (A\cup B,E)\) is bipartite, e.g., when matching students to courses, every pairwise stable matching is popular. In particular, popular matchings are guaranteed to exist. Our main contribution is to show that a max-size popular matching in G can be computed in linear time by the 2-level Gale–Shapley algorithm, which is an extension of the classical Gale–Shapley algorithm. We prove its correctness via linear programming. Second, we consider the problem of computing a max-size popular matching in \(G = (V,E)\) (not necessarily bipartite) when every agent has capacity 1, e.g., when matching students to dorm rooms. We show that even when G admits a stable matching, this problem is \(\mathsf {NP}\)-hard, which is in contrast to the tractability result in bipartite graphs.


Matchings under preferences Gale–Shapley algorithm Linear programming duality NP-hardness 



We thank the reviewers for their helpful comments. The first author wishes to thank Larry Samuelson for comments on the motivation for popular matchings. The second author wishes to thank David Manlove and Bruno Escoffier for asking her about popular matchings in the hospitals/residents setting.


  1. 1.
    Abraham, D.J., Irving, R.W., Kavitha, T., Mehlhorn, K.: Popular matchings. SIAM J. Comput. 37(4), 1030–1045 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Askalidis, G., Immorlica, N., Kwanashie, A., Manlove, D., Pountourakis, E.: Socially Stable matchings in the hospitals/residents problem. In: the 13th International Symposium on Algorithms and Data Structures (WADS), pp. 85–96 (2013)Google Scholar
  3. 3.
    Blair, C.: The lattice structure of the set of stable matchings with multiple partners. Math. Oper. Res. 13, 619–628 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Biro, P., Irving, R.W., Manlove, D.F.: Popular matchings in the marriage and roommates problems. In: the 7th International Conference in Algorithms and Complexity (CIAC), pp. 97–108 (2010)Google Scholar
  5. 5.
    Brandl, F., Kavitha, T.: Popular matchings with multiple partners. In: the 37th Foundations of Software Technology and Theoretical Computer Science (FSTTCS) (2017)Google Scholar
  6. 6.
    Canadian Resident Matching Service. How the matching algorithm works. Web document available at
  7. 7.
    Condorcet, J.A.N.C.: Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix. Imprimerie Royale, Paris (1785)Google Scholar
  8. 8.
    Cseh, Á.: Popular matchings. In: Trends in Computational Social Choice, Edited by Ulle Endriss, COST (European Cooperation in Science and Technology), pp. 105–122 (2017)Google Scholar
  9. 9.
    Cseh, Á., Huang, C.-C., Kavitha, T.: Popular matchings with two-sided preferences and one-sided ties. In: the 42nd International Colloquium on Automata, Languages, and Programming (ICALP): Part I, pp. 367–379 (2015)Google Scholar
  10. 10.
    Cseh, Á., Kavitha, T.: Popular edges and dominant matchings. In: the 18th International Conference on Integer Programming and Combinatorial Optimization (IPCO), pp. 138–151 (2016)Google Scholar
  11. 11.
    Faenza, Y., Kavitha, T., Powers, V., Zhang, X.: Popular matchings and limits to tractability. In: The Conference is the 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, pp. 2790–2809 (2019)Google Scholar
  12. 12.
    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Month. 69(1), 9–15 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gale, D., Sotomayor, M.: Some remarks on the stable matching problem. Discrete Appl. Math. 11(3), 223–232 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gärdenfors, P.: Match making: assignments based on bilateral preferences. Behav. Sci. 20(3), 166–173 (1975)CrossRefGoogle Scholar
  15. 15.
    Gupta, S., Misra, P., Saurabh, S., Zehavi, M.: Popular matching in roommates setting is NP-hard. The Conference is the 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, pp. 2810–2822 (2019)Google Scholar
  16. 16.
    Gusfield, D., Irving, R.W.: The Stable Marriage Problem: Structure and Algorithms. MIT Press, Boston (1989)zbMATHGoogle Scholar
  17. 17.
    Hamada, K., Iwama, K., Miyazaki, S.: The hospitals/residents problem with lower quotas. Algorithmica 74(1), 440–465 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hirakawa, M., Yamauchi, Y., Kijima, S., Yamashita, M.: On the structure of popular matchings in the stable marriage problem—who can join a popular matching? In: The 3rd International Workshop on Matching Under Preferences (MATCH-UP) (2015)Google Scholar
  19. 19.
    Huang, C.-C.: Classified stable matching. In: the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1235–1253 (2010)Google Scholar
  20. 20.
    Huang, C.-C., Kavitha, T.: Popular matchings in the stable marriage problem. Inf. Comput. 222, 180–194 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Huang, C.-C., Kavitha, T.: Popularity, self-duality, and mixed matchings. In: the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 2294–2310 (2017)Google Scholar
  22. 22.
    Irving, R.W.: An efficient algorithm for the stable roommates problem. J. Algorithms 6, 577–595 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Irving, R.W., Manlove, D.F., Scott, S.: The hospitals/residents problem with ties. In: the 7th Scandinavian Workshop on Algorithm Theory (SWAT), pp. 259–271 (2000)Google Scholar
  24. 24.
    Irving, R.W., Manlove, D.F., Scott, S.: Strong stability in the hospitals/residents problem. In the 20th Annual Symposium on Theoretical Aspects of Computer Science (STACS), pp. 439–450 (2003)Google Scholar
  25. 25.
    Kavitha, T.: A size-popularity tradeoff in the stable marriage problem. SIAM J. Comput. 43(1), 52–71 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kavitha, T.: Popular half-integral matchings. In: the 43rd International Colloquium on Automata, Languages, and Programming (ICALP), pp. 22.1–22.13 (2016)Google Scholar
  27. 27.
    Kavitha, T.: Max-size popular matchings and extensions.
  28. 28.
    Kavitha, T., Mestre, J., Nasre, M.: Popular mixed matchings. Theor. Comput. Sci. 412(24), 2679–2690 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Király, T., Mészáros-Karkus, Zs: Finding strongly popular b-matchings in bipartite graphs. Electron. Notes Discrete Math. 61, 735–741 (2017)CrossRefzbMATHGoogle Scholar
  30. 30.
    Manlove, D.F.: Algorithmics of Matching Under Preferences. World Scientific Publishing Company Incorporated, Singapore (2013)CrossRefzbMATHGoogle Scholar
  31. 31.
    Manlove, D.F.: The Hospitals/Residents Problem. Encyclopedia of Algorithms, pp. 926–930. Springer, Berlin (2015)Google Scholar
  32. 32.
    Nasre, M., Rawat, A.: Popularity in the generalized hospital residents setting. In: the 12th International Computer Science Symposium in Russia (CSR), pp. 245–259 (2017)Google Scholar
  33. 33.
    National Resident Matching Program. Why the Match? Web document available at
  34. 34.
    Roth, A.E.: The evolution of the labor market for medical interns and residents: a case study in game theory. J. Polit. Econ. 92(6), 991–1016 (1984)CrossRefGoogle Scholar
  35. 35.
    Roth, A.E.: Stability and polarization of interest in job matching. Econometrica 53, 47–57 (1984)CrossRefzbMATHGoogle Scholar
  36. 36.
    Roth, A.E.: On the allocation of residents to rural hospitals: a general property of two-sided matching markets. Econometrica 54(2), 425–427 (1986)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Sotomayor, M.: Three remarks on the many-to-many stable matching problem. Math. Soc. Sci. 38, 55–70 (1999)CrossRefzbMATHGoogle Scholar
  38. 38.
    Subramanian, A.: A new approach to stable matching problems. SIAM J. Comput. 23(4), 671–700 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Teo, C.-P., Sethuraman, J.: The geometry of fractional stable matchings and its applications. Math. Oper. Res. 23(4), 874–891 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Technische Universität MünchenMunichGermany
  2. 2.Tata Institute of Fundamental ResearchMumbaiIndia

Personalised recommendations