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Popular Matching in Roommates Setting Is NP-hard

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Published:26 March 2021Publication History
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Abstract

An input to the POPULAR MATCHING problem, in the roommates setting (as opposed to the marriage setting), consists of a graph G (not necessarily bipartite) where each vertex ranks its neighbors in strict order, known as its preference. In the POPULAR MATCHING problem the objective is to test whether there exists a matching M* such that there is no matching M where more vertices prefer their matched status in M (in terms of their preferences) over their matched status in M*. In this article, we settle the computational complexity of the POPULAR MATCHING problem in the roommates setting by showing that the problem is NP-complete. Thus, we resolve an open question that has been repeatedly and explicitly asked over the last decade.

References

  1. D. J. Abraham, R. W. Irving, T. Kavitha, and K. Mehlhorn. 2005. Popular matchings. In Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms (SODA’05). ACM-SIAM, 424--432. https://dl.acm.org/doi/10.5555/1070432.1070491.Google ScholarGoogle Scholar
  2. David J. Abraham, Robert W. Irving, Telikepalli Kavitha, and Kurt Mehlhorn. 2007. Popular matchings. SIAM J. Comput. 37, 4 (2007), 1030--1045. DOI:https://doi.org/10.1137/06067328XGoogle ScholarGoogle ScholarDigital LibraryDigital Library
  3. Péter Biró, Robert W. Irving, and David F. Manlove. 2010. Popular matchings in the marriage and roommates problems. In Algorithms and Complexity, Tiziana Calamoneri and Josep Diaz (Eds.). Springer, Berlin, 97--108.Google ScholarGoogle Scholar
  4. K. S. Chung. 2000. On the existence of stable roommate matchings. Games Econ. Behav. 33, 2 (2000), 206--230.Google ScholarGoogle ScholarCross RefCross Ref
  5. Agnes Cseh. 2015. Popular matchings. (2015). Combinatorial Optimization, Hausdorff Trimester Program. Retrieved from http://www.him.uni-bonn.de/combinatorial-optimization-2015/.Google ScholarGoogle Scholar
  6. Ágnes Cseh. 2017. Popular matchings. Trends Comput. Soc. Choice 105, 3 (2017), 105--122.Google ScholarGoogle Scholar
  7. Ágnes Cseh, Chien-Chung Huang, and Telikepalli Kavitha. 2017. Popular matchings with two-sided preferences and one-sided ties. SIAM J. Discr. Math. 31, 4 (2017), 2348--2377.Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Ágnes Cseh and Telikepalli Kavitha. 2016. Popular edges and dominant matchings. In Proceedings of the 18th International Conference Integer Programming and Combinatorial Optimization (IPCO’16). Springer, Berlin, 138--151.Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. M.-J.-A.-N. de C. (Marquis de) Condorcet. 1785. Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix. De l’Imprimerie royale, M. DCCLXXXV.Google ScholarGoogle Scholar
  10. Yuri Faenza, Telikepalli Kavitha, Vladlena Powers, and Xingyu Zhang. 2019. Popular matchings and limits to tractability. In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’19). ACM-SIAM, 2790--2809. DOI:https://doi.org/10.1137/1.9781611975482.173Google ScholarGoogle ScholarCross RefCross Ref
  11. D. Gale and L. Shapley. 1962. College admissions and the stability of marriage. Am. Math. Monthly 69, 1 (1962), 9--15.Google ScholarGoogle Scholar
  12. P. Gärdenfors. 1975. Match making: Assignments based on bilateral preferences. Behav. Sci. 20, 3 (1975), 166--173.Google ScholarGoogle ScholarCross RefCross Ref
  13. Sushmita Gupta, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi. 2019. Popular matching in roommates setting is NP-hard. In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’19). ACM-SIAM, USA, 2810--2822. DOI:https://doi.org/10.1137/1.9781611975482.174Google ScholarGoogle ScholarCross RefCross Ref
  14. Chien-Chung Huang and Telikepalli Kavitha. 2013. Near-popular matchings in the roommates problem. SIAM J. Discr. Math. 27, 1 (2013), 43--62.Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. Chien-Chung Huang and Telikepalli Kavitha. 2013. Popular matchings in the stable marriage problem. Inf. Comput. 222 (2013), 180--194.Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Chien-Chung Huang, Telikepalli Kavitha, Dimitrios Michail, and Meghana Nasre. 2008. Bounded unpopularity matchings. In Proceedings of the 11th Scandinavian Workshop on Algorithm Theory (SWAT’08). Springer, Berlin, 127--137.Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. Chien-Chung Huang and Telikepalli Kavitha. 2017. Popularity, mixed matchings, and self-duality. In Proceedings of the 2017 Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM, 2294--2310. DOI:https://doi.org/10.1137/1.9781611974782.151Google ScholarGoogle ScholarCross RefCross Ref
  18. Telikepalli Kavitha. 2014. A size-popularity tradeoff in the stable marriage problem. SIAM J. Comput. 43, 1 (2014), 52--71.Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. Telikepalli Kavitha. 2018. Max-size popular matchings and extensions. arXiv:1802.07440v1.Google ScholarGoogle Scholar
  20. Telikepalli Kavitha. 2018. The popular roommates problem. arXiv:1804.00141v2.Google ScholarGoogle Scholar
  21. Telikepalli Kavitha, Julián Mestre, and Meghana Nasre. 2011. Popular mixed matchings. Theor. Comput. Sci. 412, 24 (2011), 2679--2690.Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. Telikepalli Kavitha and Meghana Nasre. 2009. Optimal popular matchings. Discr. Appl. Math. 157, 14 (2009), 3181--3186.Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. Telikepalli Kavitha and Meghana Nasre. 2009. Popular matchings with variable job capacities. In Proceedings of the 20th International Symposium Algorithms and Computation (ISAAC’09). Springer, Berlin, 423--433.Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. Tamás Király and Zsuzsa Mészáros-Karkus. 2020. Finding strongly popular b-matchings in bipartite graphs. Eur. J. Combin. 88 (2020), 103105.Google ScholarGoogle ScholarCross RefCross Ref
  25. David Manlove. 2013. The House Allocation problem (with applications to reviewer assignment). (2013). Summer School on Matching Problems, Markets, and Mechanisms, Budapest 2013. Retrieved from http://econ.core.hu/english/res/MatchingSchool.html.Google ScholarGoogle Scholar
  26. David Manlove and Colin T. S. Sng. 2006. Popular matchings in the capacitated house allocation problem. In Proceedings of the14th Annual European Symposium (ESA’06). Springer, Berlin, 492--503.Google ScholarGoogle Scholar
  27. David F. Manlove. 2013. Algorithmics of Matching Under Preferences. Theoretical Computer Science, Vol. 2. World Scientific.Google ScholarGoogle Scholar
  28. Richard Matthew McCutchen. 2008. The least-unpopularity-factor and least-unpopularity-margin criteria for matching problems with one-sided preferences. In Proceedings of the Latin American Theoretical Informatics Symposium (LATIN’08). Springer, Berlin, 593--604.Google ScholarGoogle ScholarCross RefCross Ref
  29. Alvin Roth. 1986. On the allocation of residents to rural hospitals: A general property of two-sided matching markets. Econometrica 54, 2 (1986), 425—427.Google ScholarGoogle ScholarCross RefCross Ref
  30. Michael Sipser. 2006. Introduction to the Theory of Computation. Thomson Course Technology.Google ScholarGoogle Scholar
  31. Egres Open Wiki. 2019. Deciding the existence of popular matchings. Retrieved from http://lemon.cs.elte.hu/egres/open/Deciding_the_existence_of_popular_matchings.Google ScholarGoogle Scholar
  32. Y. Xiao, Z. Han, C. Yuen, and L. A. DaSilva. 2016. Carrier aggregation between operators in next generation cellular networks: A stable roommate market. IEEE Trans. Wireless Commun. 15, 1 (2016), 633--650. DOI:https://doi.org/10.1109/TWC.2015.2477077Google ScholarGoogle ScholarDigital LibraryDigital Library

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            cover image ACM Transactions on Computation Theory
            ACM Transactions on Computation Theory  Volume 13, Issue 2
            June 2021
            144 pages
            ISSN:1942-3454
            EISSN:1942-3462
            DOI:10.1145/3450495
            Issue’s Table of Contents

            Copyright © 2021 ACM

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            Association for Computing Machinery

            New York, NY, United States

            Publication History

            • Published: 26 March 2021
            • Accepted: 1 January 2021
            • Revised: 1 November 2020
            • Received: 1 December 2019
            Published in toct Volume 13, Issue 2

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