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Improved Approximation of the Stable Marriage Problem

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Algorithms - ESA 2003 (ESA 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2832))

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Abstract

The stable marriage problem has recently been studied in its general setting, where both ties and incomplete lists are allowed. It is NP-hard to find a stable matching of maximum size, while any stable matching is a maximal matching and thus trivially a factor two approximation.

In this paper, we give the first nontrivial result for approximation of factor less than two. Our algorithm achieves an approximation ratio of 2/(1+L − 2) for instances in which only men have ties of length at most L. When both men and women are allowed to have ties, we show a ratio of 13/7(< 1.858) for the case when ties are of length two. We also improve the lower bound on the approximation ratio to \(\frac{21}{19}\)(> 1.1052).

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References

  1. Bansal, V., Agrawal, A., Malhotra, V.: Stable marriages with multiple partners: efficient search for an optimal solution. In: Proc. ICALP (2003) (to appear)

    Google Scholar 

  2. Dilworth, R.P.: A Decomposition Theorem for Partially Ordered Sets. Ann. Math. 51, 161–166 (1950)

    Article  MathSciNet  Google Scholar 

  3. Dinur, I., Safra, S.: The importance of being biased. In: Proc. of 34th STOC, pp. 33–42 (2002)

    Google Scholar 

  4. Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Amer. Math. Monthly 69, 9–15 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gale, D., Sotomayor, M.: Some remarks on the stable matching problem. Discrete Applied Mathematics 11, 223–232 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gusfield, D., Irving, R.W.: The Stable Marriage Problem: Structure and Algorithms. MIT Press, Boston (1989)

    MATH  Google Scholar 

  7. Halldórsson, M., Irwing, R.W., Iwama, K., Manlove, D.F., Miyazaki, S., Morita, Y., Scott, S.: Approximability Results for Stable Marriage Problems with Ties. Theoretical Computer Science (to appear)

    Google Scholar 

  8. Halldórsson, M., Iwama, K., Miyazaki, S., Morita, Y.: Inapproximability results on stable marriage problems. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 554–568. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  9. Halldórsson, M., Iwama, K., Miyazaki, S., Yanagisawa, H.: Randomized approximation of the stable marriage problem. In: Proc. COCOON 2003 (2003) (to appear)

    Google Scholar 

  10. Halperin, E.: Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs. In: Proc. 11th SODA, pp. 329–337 (2000)

    Google Scholar 

  11. Irving, R.W.: Stable marriage and indifference. Discrete Applied Mathematics 48, 261–272 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. Irving, R.W.: Matching medical students to pairs of hospitals: a new variation on an old theme. In: Bilardi, G., Pietracaprina, A., Italiano, G.F., Pucci, G. (eds.) ESA 1998. LNCS, vol. 1461, pp. 381–392. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  13. Irving, R.W., Manlove, D.F., Scott, S.: Strong Stability in the Hospitals/Residents Problem. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 439–450. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  14. Iwama, K., Manlove, D., Miyazaki, S., Morita, Y.: Stable marriage with incomplete lists and ties. In: Wiedermann, J., Van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 443–452. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  15. Manlove, D., Irving, R.W., Iwama, K., Miyazaki, S., Morita, Y.: Hard variants of stable marriage. Theoretical Computer Science 276(1-2), 261–279 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Monien, B., Speckenmeyer, E.: Ramsey numbers and an approximation algorithm for the vertex cover problem. Acta Inf. 22, 115–123 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nemhauser, G.L., Trotter, L.E.: Vertex packing: structural properties and algorithms. Mathematical Programming 8, 232–248 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  18. Teo, C.P., Sethuraman, J.V., Tan, W.P.: Gale-Shapley Stable Marriage Problem Revisited: Strategic Issues and Applications. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.J. (eds.) IPCO 1999. LNCS, vol. 1610, pp. 429–438. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  19. Yannakakis, M., Gavril, F.: Edge dominating sets in graphs. SIAM J. Appl. Math. 38, 364–372 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zito, M.: Small maximal matchings in random graphs. In: Gonnet, G.H., Viola, A. (eds.) LATIN 2000. LNCS, vol. 1776, pp. 18–27. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Iceland, Iceland

    Magnús M. Halldórsson

  2. Graduate School of Informatics, Kyoto University, Japan

    Kazuo Iwama & Hiroki Yanagisawa

  3. Academic Center for Computing and Media Studies, Kyoto University, Japan

    Shuichi Miyazaki

Authors
  1. Magnús M. Halldórsson
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  2. Kazuo Iwama
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  3. Shuichi Miyazaki
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  4. Hiroki Yanagisawa
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Editor information

Editors and Affiliations

  1. Dipartimento di Informatica e Automazione, Università degli Studi “Roma Tre”, via della Vasca Navale 79, 00146, Rome, Italy

    Giuseppe Di Battista

  2. School of Computer Science, Tel Aviv University, 69978, Tel Aviv, Israel

    Uri Zwick

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Halldórsson, M.M., Iwama, K., Miyazaki, S., Yanagisawa, H. (2003). Improved Approximation of the Stable Marriage Problem. In: Di Battista, G., Zwick, U. (eds) Algorithms - ESA 2003. ESA 2003. Lecture Notes in Computer Science, vol 2832. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39658-1_26

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  • DOI: https://doi.org/10.1007/978-3-540-39658-1_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-20064-2

  • Online ISBN: 978-3-540-39658-1

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