Stable Marriage with General Preferences

Extended Abstract
  • Linda Farczadi
  • Konstantinos Georgiou
  • Jochen Könemann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8768)

Abstract

We propose a generalization of the classical stable marriage problem. In our model, the preferences on one side of the partition are given in terms of arbitrary binary relations, which need not be transitive nor acyclic. This generalization is practically well-motivated, and as we show, encompasses the well studied hard variant of stable marriage where preferences are allowed to have ties and to be incomplete. As a result, we prove that deciding the existence of a stable matching in our model is NP-complete. Complementing this negative result we present a polynomial-time algorithm for the above decision problem in a significant class of instances where the preferences are asymmetric. We also present a linear programming formulation whose feasibility fully characterizes the existence of stable matchings in this special case. Finally, we use our model to study a long standing open problem regarding the existence of cyclic 3D stable matchings. In particular, we prove that the problem of deciding whether a fixed 2D perfect matching can be extended to a 3D stable matching is NP- complete, showing this way that a natural attempt to resolve the existence (or not) of 3D stable matchings is bound to fail.

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References

  1. 1.
    Birnbaum, M., Schmidt, U.: An experimental investigation of violations of transitivity in choice under uncertainty. J. Risk Uncertain 37(1), 77–91 (2008)CrossRefMATHGoogle Scholar
  2. 2.
    Biró, P., McDermid, E.: Three-sided stable matchings with cyclic preferences. Algorithmica 58(1), 5–18 (2010)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Brams, S.J., Gehrlein, W.V., Roberts, F.S.: The Mathematics of Preference, Choice and Order. Springer (2009)Google Scholar
  4. 4.
    Eriksson, K., Sjöstrand, J., Strimling, P.: Three-dimensional stable matching with cyclic preferences. Math. Soc. Sci. 52(1), 77–87 (2006)CrossRefMATHGoogle Scholar
  5. 5.
    Farczadi, L., Georgiou, K., Könemann, J.: Stable marriage with general preferences. arXiv preprint arXiv:1407.1853 (2014)Google Scholar
  6. 6.
    Fishburn, P.C.: Nontransitive preferences in decision theory. J. Risk Uncertain 4(2), 113–134 (1991)CrossRefMATHGoogle Scholar
  7. 7.
    Fishburn, P.C.: Preference structures and their numerical representations. Theoretical Computer Science 217(2), 359–383 (1999)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Amer. Math. Monthly, 9–15 (1962)Google Scholar
  9. 9.
    Gale, D., Sotomayor, M.: Some remarks on the stable matching problem. Discrete Appl. Math. 11(3), 223–232 (1985)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Gusfield, D., Irving, R.W.: The stable marriage problem: structure and algorithms, vol. 54. MIT Press (1989)Google Scholar
  11. 11.
    Halldórsson, M.M., Iwama, K., Miyazaki, S., Yanagisawa, H.: Improved approximation results for the stable marriage problem. TALG 3(3), 30 (2007)CrossRefGoogle Scholar
  12. 12.
    Huang, C.C.: Circular stable matching and 3-way kidney transplant. Algorithmica 58(1), 137–150 (2010)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Irving, R.W.: Stable marriage and indifference. Discrete Appl. Math. 48(3), 261–272 (1994)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Iwama, K., Miyazaki, S., Yanagisawa, H.: A 25/17-approximation algorithm for the stable marriage problem with one-sided ties. Algorithmica 68(3), 758–775 (2014)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Knuth, D.: Stable marriage and its relation to other combinatorial problems: An introduction to the mathematical analysis of algorithms. Amer. Math. Soc., Providence (1997)Google Scholar
  16. 16.
    Manlove, D.F.: The structure of stable marriage with indifference. Discrete Appl. Math. 122(1), 167–181 (2002)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Manlove, D.F.: Algorithmics of matching under preferences. World Sci. Publishing (2013)Google Scholar
  18. 18.
    Manlove, D.F., Irving, R.W., Iwama, K., Miyazaki, S., Morita, Y.: Hard variants of stable marriage. Theor. Comput. Sci. 276(1), 261–279 (2002)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    May, K.O.: Intransitivity, utility, and the aggregation of preference patterns. Econometrica, 1–13 (1954)Google Scholar
  20. 20.
    McDermid, E.: A 3/2-approximation algorithm for general stable marriage. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 689–700. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  21. 21.
    Roth, A.E., Sotomayor, M.: Two-sided matching: A study in game- theoretic modeling and analysis, vol. 18. Cambridge University Press (1992)Google Scholar
  22. 22.
    Vande Vate, J.H.: Linear programming brings marital bliss. Oper. Res. Lett. 8(3), 147–153 (1989)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Yanagisawa, H.: Approximation algorithms for stable marriage problems. Ph.D. thesis, Citeseer (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Linda Farczadi
    • 1
  • Konstantinos Georgiou
    • 1
  • Jochen Könemann
    • 1
  1. 1.University of WaterlooWaterlooCanada

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