Document Open Access Logo

Popular Roommates in Simply Exponential Time

Author Telikepalli Kavitha

PDF

File

Thumbnail PDF
PDF
LIPIcs.FSTTCS.2019.20.pdf
  • Filesize: 0.49 MB
  • 15 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Roommates instance
  • Popular matching
  • Stable matching
  • Dual certificate

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We consider the popular matching problem in a graph G = (V,E) on n vertices with strict preferences. A matching M is popular if there is no matching N in G such that vertices that prefer N to M outnumber those that prefer M to N. It is known that it is NP-hard to decide if G has a popular matching or not. There is no faster algorithm known for this problem than the brute force algorithm that could take n! time. Here we show a simply exponential time algorithm for this problem, i.e., one that runs in O^*(k^n) time, where k is a constant.
We use the recent breakthrough result on the maximum number of stable matchings possible in such instances to analyze our algorithm for the popular matching problem. We identify a natural (also, hard) subclass of popular matchings called truly popular matchings and show an O^*(2^n) time algorithm for the truly popular matching problem.

Cite As Get BibTex

Telikepalli Kavitha. Popular Roommates in Simply Exponential Time. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.FSTTCS.2019.20

Author Details

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

Acknowledgements

Work done while visiting Max-Planck-Institut für Informatik, Saarland Informatics Campus, Germany. Thanks to Neeldhara Misra for asking me about fast exponential time algorithms for the popular roommates problem.

References

  1. D. J. Abraham, R. W. Irving, T. Kavitha, and K. Mehlhorn. Popular Matchings. SIAM Journal on Computing, 37(4):1030-1045, 2007. Google Scholar
  2. P. Biró, R. W. Irving, and D. F. Manlove. Popular matchings in the marriage and roommates problems. In Proceedings of the 7th International Conference on Algorithms and Complexity (CIAC), pages 97-108, 2010. Google Scholar
  3. F. Brandl and T. Kavitha. Two Problems in Max-Size Popular Matchings. Algorithmica, 81(7):2738-2764, 2019. Google Scholar
  4. K.S. Chung. On the Existence of Stable Roommate Matchings. Games and Economic Behavior, 33(2):206-230, 2000. Google Scholar
  5. 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
  6. Condorcet method. URL: https://en.wikipedia.org/wiki/Condorcet_method.
  7. Á. Cseh, C.-C. Huang, and T. Kavitha. Popular matchings with two-sided preferences and one-sided ties. SIAM Journal on Discrete Mathematics, 31(4):2348-2377, 2017. Google Scholar
  8. Á. Cseh and T. Kavitha. Popular edges and dominant matchings. Mathematical Programming, 172(1):209-229, 2018. Google Scholar
  9. Á. Cseh and T. Kavitha. Popular Matchings in Complete Graphs. In Proceedings of the 38th Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), pages 17:1-17:14, 2018. Google Scholar
  10. W. H. Cunningham and A. B. Marsh. A primal algorithm for optimal matching. Mathematical Programming, 8:50-72, 1978. Google Scholar
  11. B. C. Dean and S. Munshi. Faster algorithms for stable allocation problems. Algorithmica, 58(1):59-81, 2010. Google Scholar
  12. J. Edmonds. Maximum matching and a polyhedron with 0,1-vertices. Journal of Research of the National Bureau of Standards B, 69B:125-130, 1965. 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. Fleiner, R. W. Irving, and D. F. Manlove. Efficient algorithms for generalised stable marriage and roommates problems. Theoretical Computer Scienc, 381:162-176, 2007. Google Scholar
  15. F. V. Fomin and D. Kratsch. Exact exponential algorithms. Springer-Verlag New York, Inc., New York, 2010. Google Scholar
  16. D. Gale and L.S. Shapley. College admissions and the stability of marriage. American Mathematical Monthly, 69(1):9-15, 1962. Google Scholar
  17. P. Gärdenfors. Match making: assignments based on bilateral preferences. Behavioural Science, 20(3):166-173, 1975. Google Scholar
  18. S. Gupta, P. Misra, S. Saurabh, and M. Zehavi. Popular matching in roommates setting is NP-hard. In Proceedings of the 30th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2810-2822, 2019. Google Scholar
  19. D. Gusfield and R. W. Irving. The Stable Marriage Problem: Structure and Algorithms. MIT Press, Boston, MA, 1989. Google Scholar
  20. M. Hirakawa, Y. Yamauchi, S. Kijima, and M. Yamashita. 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
  21. C.-C. Huang and T. Kavitha. Near-popular matchings in the Roommates problem. SIAM Journal on Discrete Mathematics, 27(1):43-62, 2013. Google Scholar
  22. C.-C. Huang and T. Kavitha. Popular matchings in the stable marriage problem. Information and Computation, 222:180-194, 2013. Google Scholar
  23. C.-C. Huang and T. Kavitha. Popularity, mixed matchings, and self-duality. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2294-2310, 2017. Google Scholar
  24. R.W. Irving. An efficient algorithm for the stable roommates problem. Journal of Algorithms, 6:577-595, 1985. Google Scholar
  25. A. R. Karlin, S. Oveis Gharan, and R. Weber. A simply exponential upper bound on the maximum number of stable matchings. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 920-925, 2018. Google Scholar
  26. T. Kavitha. A size-popularity tradeoff in the stable marriage problem. SIAM Journal on Computing, 43(1):52-71, 2014. Google Scholar
  27. 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
  28. T. Kavitha, J. Mestre, and M. Nasre. Popular Mixed Matchings. Theoretical Computer Science, 412(24):2679-2690, 2011. Google Scholar
  29. E. McDermid and R. W. Irving. Sex-equal stable matchings: Complexity and exact algorithms. Algorithmica, 68:545-570, 2014. Google Scholar
  30. A. Subramanian. A New Approach to Stable Matching Problems. SIAM Journal on Computing, 23(4):671-700, 1994. 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
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact