Maximum Matchings and Popularity

Author Telikepalli Kavitha

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Telikepalli Kavitha
  • Tata Institute of Fundamental Research, Mumbai, India


Thanks to Yuri Faenza and the reviewers for helpful comments and suggestions on the presentation.

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Telikepalli Kavitha. Maximum Matchings and Popularity. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 85:1-85:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Let G be a bipartite graph where every node has a strict ranking of its neighbors. For any node, its preferences over neighbors extend naturally to preferences over matchings. A maximum matching M in G is a popular max-matching if for any maximum matching N in G, the number of nodes that prefer M is at least the number that prefer N. Popular max-matchings always exist in G and they are relevant in applications where the size of the matching is of higher priority than node preferences. Here we assume there is also a cost function on the edge set. So what we seek is a min-cost popular max-matching. Our main result is that such a matching can be computed in polynomial time. We show a compact extended formulation for the popular max-matching polytope and the algorithmic result follows from this. In contrast, it is known that the popular matching polytope has near-exponential extension complexity and finding a min-cost popular matching is NP-hard.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Bipartite graphs
  • Popular matchings
  • Stable matchings
  • Dual certificates


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  1. A. Abdulkadiroğlu and T. Sönmez. School choice: a mechanism design approach. American Economic Review, 93(3):729-747, 2003. Google Scholar
  2. D. J. Abraham, K. Cechlŕová, D. F. Manlove, and K. Mehlhorn. Pareto optimality in house allocation problems. In Proceedings of the 15th International Symposium on Algorithms and Computation (ISAAC), pages 3-15, 2004. Google Scholar
  3. T. Barnagarwala. 1172 private doctors sign up to treat patients at govt hospitals in Mumbai. Indian Express, May 11, 2020. Google Scholar
  4. S. Baswana, P. P. Chakrabarti, S. Chandran, Y. Kanoria, and U. Patange. Centralized admissions for engineering colleges in India. INFORMS Journal on Applied Analytics, 49(5):338-354, 2019. Google Scholar
  5. P. Biro, D. F. Manlove, and S. Mittal. Size versus stability in the marriage problem. Theoretical Computer Science, 411:1828-1841, 2010. Google Scholar
  6. Canadian Resident Matching Service. How the matching algorithm works. URL:
  7. M.-J.-A.-N. de C. (Marquis de) Condorcet. Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix. L'Imprimerie Royale, 1785. Google Scholar
  8. Á. Cseh. Popular matchings. Trends in Computational Social Choice, Ulle Endriss (ed.), 2017. Google Scholar
  9. Á. Cseh, Y. Faenza, T. Kavitha, and V. Powers. Understanding popular matchings via stable matchings. URL:
  10. Á. Cseh and T. Kavitha. Popular edges and dominant matchings. Mathematical Programming, 172(1):209-229, 2018. Google Scholar
  11. P. Eirinakis, D. Magos, I. Mourtos, and P. Miliotis. Polyhedral aspects of stable marriage. Mathematics of Operations Research, 39(3):656-671, 2014. Google Scholar
  12. Y. Faenza and T. Kavitha. Quasi-popular matchings, optimality, and extended formulations. In Proceedings of the 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 325-344, 2020. Google Scholar
  13. Y. Faenza, T. Kavitha, V. Powers, and X. Zhang. Popular matchings and limits to tractability. In Proceedings of the 30th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2790-2809, 2019. Google Scholar
  14. T. Feder. A new fixed point approach for stable networks and stable marriages. Journal of Computer and System Sciences, 45(2):233-284, 1992. Google Scholar
  15. T. Feder. Network flow and 2-satisfiability. Algorithmica, 11(3):291-319, 1994. Google Scholar
  16. T. Fleiner. A fixed-point approach to stable matchings and some applications. Mathematics of Operations Research, 28(1):103-126, 2003. Google Scholar
  17. D. Gale and L.S. Shapley. College admissions and the stability of marriage. American Mathematical Monthly, 69(1):9-15, 1962. Google Scholar
  18. D. Gale and M. Sotomayor. Some remarks on the stable matching problem. Discrete Applied Mathematics, 11:223-232, 1985. Google Scholar
  19. P. Gärdenfors. Match making: assignments based on bilateral preferences. Behavioural Science, 20:166-173, 1975. Google Scholar
  20. C.-C. Huang and T. Kavitha. Popular matchings in the stable marriage problem. Information and Computation, 222:180-194, 2013. Google Scholar
  21. R. W. Irving, P. Leather, and D. Gusfield. An efficient algorithm for the "optimal" stable marriage. Journal of the ACM, 34(3):532-543, 1987. Google Scholar
  22. P. B. Jaiswal. Mumbai struggles to tackle increasing number of COVID-19 cases. The Week, May 6, 2020. Google Scholar
  23. T. Kavitha. A size-popularity tradeoff in the stable marriage problem. SIAM Journal on Computing, 43(1):52-71, 2014. Google Scholar
  24. T. Kavitha. Popular half-integral matchings. In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP), pages 22:1-22:13, 2016. Google Scholar
  25. T. Kavitha. Popular matchings with one-sided bias. In Proceedings of the 47th International Colloquium on Automata, Languages, and Programming (ICALP), pages 70:1-70:18, 2020. Google Scholar
  26. M. McCutchen. The least-unpopularity-factor and least-unpopularity-margin criteria for matching problems with one-sided preferences. In Proceedings of the 8th Latin American Symposium on Theoretical Informatics (LATIN), pages 593-604, 2008. Google Scholar
  27. S. Merrill and B. Grofman. A unified theory of voting: directional and proximity spatial models. Cambridge University Press, 1999. Google Scholar
  28. National Resident Matching Program. Why the Match? URL:
  29. P. A. Robards. Applying the two-sided matching processes to the United States Navy enlisted assignment process. Master’s Thesis, Naval Postgraduate School, Monterey, Canada, 2001. Google Scholar
  30. U. G. Rothblum. Characterization of stable matchings as extreme points of a polytope. Mathematical Programming, 54:57-67, 1992. Google Scholar
  31. C.-P. Teo and J. Sethuraman. The geometry of fractional stable matchings and its applications. Mathematics of Operations Research, 23(4):874-891, 1998. Google Scholar
  32. E. G. Thurber. Concerning the maximum number of stable matchings in the stable marriage problem. Discrete Mathematics, 248(1-3):195-219, 2002. Google Scholar
  33. J. H. Vande Vate. Linear programming brings marital bliss. Operations Research Letters, 8(3):147-153, 1989. Google Scholar
  34. W. Yang, J. A. Giampapa, and K. Sycara. Two-sided matching for the US Navy detailing process with market complication. Technical Report CMU-R1-TR-03-49, Robotics Institute, Carnegie Mellon University, 2003. Google Scholar
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