Size Versus Stability in the Marriage Problem

  • Péter Biró
  • David F. Manlove
  • Shubham Mittal
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5426)


Given an instance I of the classical Stable Marriage problem with Incomplete preference lists (smi), a maximum cardinality matching can be larger than a stable matching. In many large-scale applications of smi, we seek to match as many agents as possible. This motivates the problem of finding a maximum cardinality matching in I that admits the smallest number of blocking pairs (so is “as stable as possible”). We show that this problem is NP-hard and not approximable within n 1 − ε , for any ε> 0, unless P=NP, where n is the number of men in I. Further, even if all preference lists are of length at most 3, we show that the problem remains NP-hard and not approximable within δ, for some δ> 1. By contrast, we give a polynomial-time algorithm for the case where the preference lists of one sex are of length at most 2.


Perfect Match Vertex Cover Maximum Match Stable Match Preference List 
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.


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  1. 1.
    Abdulkadiroǧluand, A., Sönmez, T.: School choice: A mechanism design approach. American Economic Review 93(3), 729–747 (2003)CrossRefGoogle Scholar
  2. 2.
    Abraham, D.J., Biró, P., Manlove, D.F.: “Almost stable” matchings in the roommates problem. In: Erlebach, T., Persinao, G. (eds.) WAOA 2005. LNCS, vol. 3879, pp. 1–14. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Abraham, D.J., Irving, R.W., Manlove, D.F.: Two algorithms for the Student-Project allocation problem. Journal of Discrete Algorithms 5(1), 79–91 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Berman, P., Karpinski, M., Scott, A.D.: Scott Approximation hardness of short symmetric instances of MAX-3SAT. Electronic Colloquium on Computational Complexity Report, number 49 (2003)Google Scholar
  5. 5.
    Biró, P., Manlove, D.F., Mittal, S.: Size versus stability in the marriage problem. Technical Report TR-2008-283, University of Glasgow, Department of Computing Science (2008)Google Scholar
  6. 6.
    Eriksson, K., Häggström, O.: Instability of matchings in decentralized markets with various preference structures. International Journal of Game Theory (2008)Google Scholar
  7. 7.
    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. American Mathematical Monthly 69, 9–15 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gale, D., Sotomayor, M.: Some remarks on the stable matching problem. Discrete Applied Mathematics 11, 223–232 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gusfield, D., Irving, R.W.: The Stable Marriage Problem: Structure and Algorithms. MIT Press, Cambridge (1989)zbMATHGoogle Scholar
  10. 10.
    Irving, R.W.: An efficient algorithm for the “stable roommates” problem. Journal of Algorithms 6, 577–595 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Irving, R.W.: Matching medical students to pairs of hospitals: A new variation on a well-known theme. In: Bilardi, G., Pietracaprina, A., Italiano, G.F., Pucci, G. (eds.) ESA 1998. LNCS, vol. 1461, pp. 381–392. Springer, Heidelberg (1998)Google Scholar
  12. 12.
    Irving, R.W., Leather, P.: The complexity of counting stable marriages. SIAM Journal on Computing 15(3), 655–667 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Irving, R.W., Manlove, D.F.: The Stable Roommates Problem with Ties. Journal of Algorithms 43, 85–105 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Khuller, S., Mitchell, S.G., Vazirani, V.V.: On-line algorithms for weighted bipartite matching and stable marriages. Theoretical Computer Science 127, 255–267 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kujansuu, E., Lindberg, T., Mäkinen, E.: The stable roommates problem and chess tournament pairings. Divulgaciones Matemáticas 7(1), 19–28 (1999)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Manlove, D.F., Irving, R.W., Iwama, K., Miyazaki, S., Morita, Y.: Hard variants of stable marriage. Theoretical Computer Science 276(1-2), 261–279 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    O’Malley, G.: Algorithmic Aspects of Stable Matching Problems. PhD thesis, University of Glasgow, Department of Computing Science (2007)Google Scholar
  18. 18.
    Robards, P.A.: Applying two-sided matching processes to the united states navy enlisted assignment process. Master’s thesis, Naval Postgraduate School, Monterey, California (2001)Google Scholar
  19. 19.
    Ronn, E.: NP-complete stable matching problems. Journal of Algorithms 11, 285–304 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Roth, A.E.: 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
  21. 21.
    Roth, A.E., Sönmez, T., Utku Ünver, M.: Kidney exchange. Quarterly Journal of Economics 119, 457–488 (2004)CrossRefzbMATHGoogle Scholar
  22. 22.
    Roth, A.E., Sönmez, T., Utku Ünver, M.: Pairwise kidney exchange. Journal of Economic Theory 125, 151–188 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Yang, W., Giampapa, J.A., Sycara, K.: Two-sided matching for the U.S. Navy Detailing Process with market complication. Technical Report CMU-RI-TR-03-49, Robotics Institute, Carnegie-Mellon University (2003)Google Scholar
  24. 24. (National Resident Matching Program website)
  25. 25. (Canadian Resident Matching Service website)
  26. 26. (Scottish Foundation Allocation Scheme website)
  27. 27. (New England Program for Kidney Exchange website)

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Péter Biró
    • 1
  • David F. Manlove
    • 1
  • Shubham Mittal
    • 2
  1. 1.Department of Computing ScienceUniversity of GlasgowGlasgowUK
  2. 2.Department of Computer Science and Engineering, Block VIIndian Institute of Technology, Delhi, Hauz KhasNew DelhiIndia

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