A Constraint Programming Approach to the Stable Marriage Problem

  • Ian P. Gent
  • Robert W. Irving
  • David F. Manlove
  • Patrick Prosser
  • Barbara M. Smith
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2239)

Abstract

The Stable Marriage problem (SM) is an extensively-studied combinatorial problem with many practical applications. In this paper we present two encodings of an instance I of SM as an instance J of a Constraint Satisfaction Problem. We prove that, in a precise sense, establishing arc consistency in J is equivalent to the action of the established Extended Gale/Shapley algorithm for SM on I. As a consequence of this, the man-optimal and woman-optimal stable matchings can be derived immediately. Furthermore we show that, in both encodings, all solutions of I may be enumerated in a failure-free manner. Our results indicate the applicability of Constraint Programming to the domain of stable matching problems in general, many of which are NP-hard.

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References

  1. 1.
    C. Bessière and J-C Régin. Arc consistency for general constraint networks: Preliminary results. In Proceedings of IJCAI’97, pages 398–404, 1997.Google Scholar
  2. 2.
    M. Davis, G. Logemann, and D. Loveland. A machine program for theoremproving. Communications of the ACM, 5:394–397, 1962.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Eugene C. Freuder. A suficient condition for backtrack-free search. Journal of the ACM, 29:24–32, 1982.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    D. Gale and L.S. Shapley. College admissions and the stability of marriage. American Mathematical Monthly, 69:9–15, 1962.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    D. Gale and M. Sotomayor. Some remarks on the stable matching problem. Discrete Applied Mathematics, 11:223–232, 1985.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    D. Gusfield and R. W. Irving. The Stable Marriage Problem: Structure and Algorithms. The MIT Press, 1989.Google Scholar
  7. 7.
    P. Jeavons, D. Cohen, and M. Gyssens. A unifying framework for tractable constraints. In Proceedings CP’95, volume LNCS 976, pages 276–291. Springer, 1995.Google Scholar
  8. 8.
    D.F. Manlove, R.W. Irving, K. Iwama, S. Miyazaki, and Y. Morita. Hard variants of stable marriage. To appear in Theoretical Computer Science.Google Scholar
  9. 9.
    C. Ng and D.S. Hirschberg. Lower bounds for the stable marriage problem and its variants. SIAM Journal on Computing, 19:71–77, 1990.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    C. Ng and D.S. Hirschberg. Three-dimensional stable matching problems. SIAM Journal on Discrete Mathematics, 4:245–252, 1991.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    E. Ronn. NP-complete stable matching problems. Journal of Algorithms, 11:285–304, 1990.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    A.E. Roth. The evolution of the labor market for medical interns and residents: a case study in game theory. Journal of Political Economy, 92(6):991–1016, 1984.CrossRefGoogle Scholar
  13. 13.
    A.E. Roth and M.A.O. Sotomayor. Two-sided matching: a study in game-theoretic modeling and analysis, volume 18 of Econometric Society Monographs. Cambridge University Press, 1990.Google Scholar
  14. 14.
    J.E. Vande Vate. Linear programming brings marital bliss. Operations Research Letters, 1989.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Ian P. Gent
    • 1
  • Robert W. Irving
    • 2
  • David F. Manlove
    • 2
  • Patrick Prosser
    • 2
  • Barbara M. Smith
    • 3
  1. 1.School of Computer ScienceUniversity of St. AndrewsScotland
  2. 2.Department of Computing ScienceUniversity of GlasgowScotland
  3. 3.School of Computing and MathematicsUniversity of HuddersfieldEngland

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