Advertisement

Stable Marriage with Groups of Similar Agents

  • Kitty Meeks
  • Baharak RastegariEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11316)

Abstract

Many important stable matching problems are known to be NP-hard, even when strong restrictions are placed on the input. In this paper we seek to identify structural properties of instances of stable matching problems which will allow us to design efficient algorithms using elementary techniques. We focus on the setting in which all agents involved in some matching problem can be partitioned into k different types, where the type of an agent determines his or her preferences, and agents have preferences over types (which may be refined by more detailed preferences within a single type). This situation would arise in practice if agents form preferences solely based on some small collection of agents’ attributes. We also consider a generalisation in which each agent may consider some small collection of other agents to be exceptional, and rank these in a way that is not consistent with their types; this could happen in practice if agents have prior contact with a small number of candidates. We show that (for the case without exceptions), the well-known NP-hard matching problem Max SMTI (that of finding the maximum cardinality stable matching in an instance of stable marriage with ties and incomplete lists) belongs to the parameterised complexity class FPT when parameterised by the number of different types of agents needed to describe the instance. This tractability result can be extended to the setting in which each agent promotes at most one “exceptional” candidate to the top of his/her list (when preferences within types are not refined), but the problem remains NP-hard if preference lists can contain two or more exceptions and the exceptional candidates can be placed anywhere in the preference lists.

Notes

Acknowledgements

The first author is supported by a Personal Research Fellowship from the Royal Society of Edinburgh (funded by the Scottish Government). Both authors are extremely grateful to David Manlove for his insightful comments on a preliminary version of this manuscript.

References

  1. 1.
    Aziz, H., de Keijzer, B.: Complexity of coalition structure generation. In: Proceedings of the 10th International Conference on Autonomous Agents and Multiagent Systems, AAMAS 2011, pp. 191–198 (2011)Google Scholar
  2. 2.
    Bhatnagar, N., Greenberg, S., Randall, D.: Sampling stable marriages: why spouse-swapping won’t work. In: Proceedings of the 19th ACM/SIAM Symposium on Discrete Algorithms, SODA 2008, pp. 1223–1232. ACM-SIAM (2008)Google Scholar
  3. 3.
    Biró, P., Manlove, D., Mittal, S.: Size versus stability in the marriage problem. Theor. Comput. Sci. 411, 1828–1841 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chebolu, P., Goldberg, L.A., Martin, R.: The complexity of approximately counting stable matchings. Theor. Comput. Sci. 437, 35–68 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chebolu, P., Goldberg, L.A., Martin, R.: The complexity of approximately counting stable roommate assignments. J. Comput. Syst. Sci. 78(5), 1579–1605 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Choo, E., Siow, A.: Who marries whom and why. J. Polit. Econ. 114(1), 175–201 (2006)CrossRefGoogle Scholar
  7. 7.
    Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-21275-3CrossRefzbMATHGoogle Scholar
  8. 8.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, London (2013).  https://doi.org/10.1007/978-1-4471-5559-1CrossRefzbMATHGoogle Scholar
  9. 9.
    Echenique, F., Lee, S., Shum, M., Yenmez, M.B.: The revealed preference theory of stable and extremal stable matchings. Econometrica 81(1), 153–171 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006).  https://doi.org/10.1007/3-540-29953-XCrossRefzbMATHGoogle Scholar
  11. 11.
    Gale, D., Shapley, L.: College admissions and the stability of marriage. Am. Math. Mon. 69, 9–15 (1962)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gupta, S., Roy, S., Saurabh, S., Zehavi, M.: Balanced stable marriage: how close is close enough. Technical report 1707.09545, CoRR, Cornell University Library (2017)Google Scholar
  13. 13.
    Gupta, S., Saurabh, S., Zehavi, M.: On treewidth and stable marriage. Technical report 1707.05404, CoRR, Cornell University Library (2017)Google Scholar
  14. 14.
    Irving, R., Manlove, D., Scott, S.: The stable marriage problem with master preference lists. Discret. Appl. Math. 156(15), 2959–2977 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Manlove, D.: Algorithmics of Matching Under Preferences. World Scientific, Singapore (2013)CrossRefGoogle Scholar
  16. 16.
    Manlove, D., Irving, R., Iwama, K., Miyazaki, S., Morita, Y.: Hard variants of stable marriage. Theor. Comput. Sci. 276(1–2), 261–279 (2002)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Marx, D., Schlotter, I.: Parameterized complexity and local search approaches for the stable marriage problem with ties. Algorithmica 58(1), 170–187 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Meeks, K., Rastegari, B.: Solving hard stable matching problems involving groups of similar agents. Technical report 1708.04109, CoRR, Cornell University Library (2018)Google Scholar
  19. 19.
    Mnich, M., Schlotter, I.: Stable marriage with covering constraints–a complete computational trichotomy. In: Bilò, V., Flammini, M. (eds.) SAGT 2017. LNCS, vol. 10504, pp. 320–332. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-66700-3_25CrossRefGoogle Scholar
  20. 20.
    O’Malley, G.: Algorithmic aspects of stable matching problems. Ph.D. thesis, Department of Computing Science, University of Glasgow (2007)Google Scholar
  21. 21.
    Orlin, J.B.: Max flows in \(\cal{O}(nm)\) time, or better. In: Proceedings of the 45th Annual ACM Symposium on Theory of Computing, STOC 2013, pp. 765–774. ACM (2013)Google Scholar
  22. 22.
    Shrot, T., Aumann, Y., Kraus, S.: On agent types in coalition formation problems. In: Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems, AAMAS 2010, pp. 757–764 (2010)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of Computing ScienceUniversity of GlasgowGlasgowUK
  2. 2.Electronics and Computer ScienceUniversity of SouthamptonSouthamptonUK

Personalised recommendations