Advertisement

The Unsplittable Stable Marriage Problem

  • Brian C. Dean
  • Michel X. Goemans
  • Nicole Immorlica
Part of the IFIP International Federation for Information Processing book series (IFIPAICT, volume 209)

Abstract

The Gale-Shapley “propose/reject” algorithm is a well-known procedure for solving the classical stable marriage problem. In this paper we study this algorithm in the context of the many-to-many stable marriage problem, also known as the stable allocation or ordinal transportation problem. We present an integral variant of the Gale-Shapley algorithm that provides a direct analog, in the context of “ordinal” assignment problems, of a well-known bicriteria approximation algorithm of Shmoys and Tardos for scheduling on unrelated parallel machines with costs. If we are assigning, say, jobs to machines, our algorithm finds an unsplit (non-preemptive) stable assignment where every job is assigned at least as well as it could be in any fractional stable assignment, and where each machine is congested by at most the processing time of the largest job.

Keywords

Integral Variant Stable Match Fractional Variant Preference List Unrelated Parallel Machine 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    M. Baiou and M. Balinski. Erratum: The stable allocation (or ordinal transportation) problem. Mathematics of Operations Research, 27(4):662–680, 2002.CrossRefMathSciNetGoogle Scholar
  2. 2.
    Y. Dinitz, N. Garg, and M.X. Goemans. On the single-source unsplittable flow problem. Combinatorica, 19:17–41, 1999.CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    D. Gale and L.S. Shapley. College admissions and the stability of marriage. American Mathematical Monthly, 69(1):9–14, 1962.CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    D. Gusfield and R. Irving. The Stable Marriage Problem: Structure and Algorithms. MIT Press, 1989.Google Scholar
  5. 5.
    N. Immorlica and M. Mahdian. Marriage, honesty, and stability. In Proceedings of 16th ACM Symposium on Discrete Algorithms, pages 53–62, 2005.Google Scholar
  6. 6.
    B. Klaus and F. Klijn. Stable matchings and preferences of couples. Journal of Economic Theory, 121:75–106, 2005.CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    J.M. Kleinberg. Approximation algorithms for disjoint paths problems. PhD thesis, M.I.T., 1996.Google Scholar
  8. 8.
    D. E. Knuth. Stable marriage and its relation to other combinatorial problems. In CRM Proceedings and Lecture Notes, vol. 10, American Mathematical Society, Providence, RI. (English translation of Marriages Stables, Les Presses de L’Université de Montréal, 1976), 1997.Google Scholar
  9. 9.
    E. Ronn. Np-complete stable matching problems. Journal of Algorithms, 11:285–304, 1990.CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    A.E. Roth. The evolution of the labor market for medical interns and residents: a case study in game theory. Journal of Political Economy, 92:991–1016, 1984.CrossRefGoogle Scholar
  11. 11.
    A.E. Roth. On the allocation of residents to rural hospitals: a general property of two-sided matching markets. Econometrica, 54:425–427, 1986.CrossRefMathSciNetGoogle Scholar
  12. 12.
    A.E. Roth. The national residency matching program as a labor market. Journal of the American Medical Association, 275(13):1054–1056, 1996.CrossRefGoogle Scholar
  13. 13.
    A.E. Roth and E. Peranson. The redesign of the matching market for american physicians: Some engineering aspects of economic design. American Economic Review, 89: 748–780, 1999.CrossRefGoogle Scholar
  14. 14.
    A.E. Roth and M. Sotomayor. Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis. Cambridge University Press, 1990.Google Scholar
  15. 15.
    D.B. Shmoys and ÉE. Tardos. Scheduling unrelated machines with costs. In Proceedings of the 4th annual ACM-SIAM Symposium on Discrete algorithms (SODA), pages 448–454, 1993.Google Scholar
  16. 16.
    M. Skutella. Approximating the single source unsplittable min-cost flow problem. In Proceedings of the 41st Annual Symposium on Foundations of Computer Science (FOCS), pages 136–145, 2000.Google Scholar

Copyright information

© International Federation for Information Processing 2006

Authors and Affiliations

  • Brian C. Dean
    • 1
  • Michel X. Goemans
    • 2
  • Nicole Immorlica
    • 3
  1. 1.Department of Computer ScienceClemson UniversityClemsonUSA
  2. 2.Department of MahematicsM.I.TUSA
  3. 3.Microsoft ResearchUSA

Personalised recommendations