Advertisement

Algorithmica

, Volume 51, Issue 3, pp 342–356 | Cite as

A ( \(2-c\frac{1}{\sqrt{N}}\) )-Approximation Algorithm for the Stable Marriage Problem

  • Kazuo Iwama
  • Shuichi MiyazakiEmail author
  • Naoya Yamauchi
Article

Abstract

We consider the problem of finding a stable matching of maximum size when both ties and unacceptable partners are allowed in preference lists. This problem is NP-hard and the current best known approximation algorithm achieves the approximation ratio 2−c(log N)/N, where c is an arbitrary positive constant and N is the number of men in an input. In this paper, we improve the ratio to \(2-c/\sqrt{N}\) , where c is an arbitrary constant that satisfies \(c\leq 1/{(4\sqrt{6})}\) .

Keywords

The stable marriage problem Ties Incomplete lists Approximation algorithms 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abraham, D.J., Cechlárová, K., Manlove, D.F., Mehlhorn, K.: Pareto optimality in house allocation problems. In: Proceedings of ISAAC 2004. Lecture Notes in Computer Science, vol. 3341, pp. 3–15. Springer, Berlin (2004) Google Scholar
  2. 2.
    Abraham, D.J., Irving, R.W., Kavitha, T., Mehlhorn, K.: Popular matchings. In: Proceedings of SODA 2005, pp. 424–432 (2005) Google Scholar
  3. 3.
    Bansal, V., Agrawal, A., Malhotra, V.: Stable marriages with multiple partners: efficient search for an optimal solution. In: Proceedings of ICALP 2003. Lecture Notes in Computer Science, vol. 2719, pp. 527–542. Springer, Berlin (2003) Google Scholar
  4. 4.
    Bar-Yehuda, R., Even, S.: A local-ratio theorem for approximating the weighted vertex cover problem. In: Analysis and Design of Algorithms for Combinatorial Problems. Annals of Discrete Mathematics, vol. 25, pp. 27–46. Elsevier Science, Amsterdam (1985) CrossRefGoogle Scholar
  5. 5.
    Berman, P., Fujito, T.: On the approximation properties of independent set problem in degree 3 graphs. In: Proceedings of WADS 95. Lecture Notes in Computer Science, vol. 955, pp. 449–460. Springer, Berlin (1995) Google Scholar
  6. 6.
    Canadian Resident Matching Service (CaRMS): http://www.carms.ca/
  7. 7.
    Cechlárová, K.: On the complexity of exchange-stable roommates. Discrete Appl. Math. 116, 279–287 (2002) CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Fleiner, T.: A fixed-point approach to stable matchings and some applications. Math. Oper. Res. 28(1), 103–126 (2003) CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Mon. 69, 9–15 (1962) CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Gale, D., Sotomayor, M.: Some remarks on the stable matching problem. Discrete Appl. Math. 11, 223–232 (1985) CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Gusfield, D., Irving, R.W.: The Stable Marriage Problem: Structure and Algorithms. MIT Press, Boston (1989) zbMATHGoogle Scholar
  12. 12.
    Halldórsson, M.M., Irving, R.W., Iwama, K., Manlove, D.F., Miyazaki, S., Morita, Y., Scott, S.: Approximability results for stable marriage problems with ties. Theor. Comput. Sci. 306, 431–447 (2003) CrossRefzbMATHGoogle Scholar
  13. 13.
    Halldórsson, M.M., Iwama, K., Miyazaki, S., Yanagisawa, H.: Improved approximation of the stable marriage problem. In: Proceedings of ESA 2003. Lecture Notes in Computer Science, vol. 2832, pp. 266–277. Springer, Berlin (2003) Google Scholar
  14. 14.
    Halldórsson, M.M., Iwama, K., Miyazaki, S., Yanagisawa, H.: Randomized approximation of the stable marriage problem. Theor. Comput. Sci. 325(3), 439–465 (2004) CrossRefzbMATHGoogle Scholar
  15. 15.
    Halperin, E.: Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs. In: Proceedings of SODA 2000, pp. 329–337 (2000) Google Scholar
  16. 16.
    Irving, R.W.: Stable marriage and indifference. Discrete Appl. Math. 48, 261–272 (1994) CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Irving, R.W.: Matching medical students to pairs of hospitals: a new variation on a well-known theme. In: Proceedings of ESA 98. Lecture Notes in Computer Science, vol. 1461, pp. 381–392. Springer, Berlin (1998) Google Scholar
  18. 18.
    Irving, R.W., Manlove, D.F., Scott, S.: The hospital/residents problem with ties. In: Proceedings of SWAT 2000. Lecture Notes in Computer Science, vol. 1851, pp. 259–271. Springer, Berlin (2000) CrossRefGoogle Scholar
  19. 19.
    Irving, R.W., Kavitha, T., Mehlhorn, K., Michail, D., Paluch, K.: Rank-maximal matchings. In: Proceedings of SODA 2004, pp. 68–75 (2004) Google Scholar
  20. 20.
    Irving, R.W., Manlove, D.F., Scott, S.: Strong stability in the hospitals/residents problem. In: Proceedings of STACS 2003. Lecture Notes in Computer Science, vol. 2607, pp. 439–450. Springer, Berlin (2003) Google Scholar
  21. 21.
    Iwama, K., Manlove, D.F., Miyazaki, S., Morita, Y.: Stable marriage with incomplete lists and ties. In: Proceedings of ICALP 99. Lecture Notes in Computer Science, vol. 1644, pp. 443–452. Springer, Berlin (1999) Google Scholar
  22. 22.
    Iwama, K., Miyazaki, S., Okamoto, K.: A (2−clog N/N)-approximation algorithm for the stable marriage problem. In: Proceedings of SWAT 2004. Lecture Notes in Computer Science, vol. 3111, pp. 349–361. Springer, Berlin (2004) Google Scholar
  23. 23.
    Japanese Resident Matching Program (JRMP): http://www.jrmp.jp/
  24. 24.
    Karakostas, G.: A better approximation ratio for the Vertex Cover problem. ECCC Report, TR04-084 (2004) Google Scholar
  25. 25.
    Karpinski, M.: Polynomial time approximation schemes for some dense instances of NP-hard optimization problems. In: Proceedings of RANDOM 97. Lecture Notes in Computer Science, vol. 1269, pp. 1–14. Springer, Berlin (1997) Google Scholar
  26. 26.
    Kavitha, T., Mehlhorn, K., Michail, D., Paluch, K.: Strongly stable matchings in time O(nm) and extension to the hospitals-residents problem. In: Proceedings of STACS 2004, pp. 222–233 (2004) Google Scholar
  27. 27.
    Lawler, E.L.: Combinatorial Optimization: Networks and Matroids. Rinehart and Winston, Holt (1976) zbMATHGoogle Scholar
  28. 28.
    Le, T., Bhushan, V., Amin, C.: First Aid for the Match, 2nd edn. McGraw-Hill, New York (2001) Google Scholar
  29. 29.
    Manlove, D.F., Irving, R.W., Iwama, K., Miyazaki, S., Morita, Y.: Hard variants of stable marriage. Theor. Comput. Sci. 276(1/2), 261–279 (2002) CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Monien, B., Speckenmeyer, E.: Ramsey numbers and an approximation algorithm for the vertex cover problem. Acta Inf. 22, 115–123 (1985) CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Nagamochi, H., Nishida, Y., Ibaraki, T.: Approximability of the minimum maximal matching problem in planar graphs. Inst. Electr. Inf. Commun. Eng. Trans. Fundam. E86-A, 3251–3258 (2003) Google Scholar
  32. 32.
    Teo, C.P., Sethuraman, J.V., Tan, W.P.: Gale–Shapley stable marriage problem revisited: strategic issues and applications. In: Proceedings of IPCO 99, pp. 429–438 (1999) Google Scholar
  33. 33.
    Zito, M.: Randomized techniques in combinatorial algorithmics, Ph.D. thesis, Department of Computer Science, University of Warwick (1999) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Graduate School of InformaticsKyoto UniversityKyotoJapan
  2. 2.Academic Center for Computing and Media StudiesKyoto UniversityKyotoJapan

Personalised recommendations