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The Stable Roommates Problem with Short Lists

  • Ágnes Cseh
  • Robert W. Irving
  • David F. Manlove
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9928)

Abstract

We consider two variants of the classical Stable Roommates problem with Incomplete (but strictly ordered) preference lists (sri) that are degree constrained, i.e., preference lists are of bounded length. The first variant, egal d-sri, involves finding an egalitarian stable matching in solvable instances of sri with preference lists of length at most d. We show that this problem is \(\textsf {NP}\)-hard even if \(d=3\). On the positive side we give a \(\frac{2d+3}{7}\)-approximation algorithm for \(d\in \{3,4,5\}\) which improves on the known bound of 2 for the unbounded preference list case. In the second variant of sri, called d-srti, preference lists can include ties and are of length at most d. We show that the problem of deciding whether an instance of d-srti admits a stable matching is \(\textsf {NP}\)-complete even if \(d=3\). We also consider the “most stable” version of this problem and prove a strong inapproximability bound for the \(d=3\) case. However for \(d=2\) we show that the latter problem can be solved in polynomial time.

Keywords

Approximation Algorithm Match History Stable Match Preference List Minimum Vertex Cover 
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.
    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
  2. 2.
    Berman, P., Karpinski, M., Scott, A.D.: Approximation hardness of short symmetric instances of MAX-3SAT. ECCC report, no. 49 (2003)Google Scholar
  3. 3.
    Biró, P., Manlove, D.F., McDermid, E.J.: “Almost stable" matchings in the roommates problem with bounded preference lists. Theor. Comput. Sci. 432, 10–20 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Biró, P., Manlove, D.F., McDermid, E.J.: Almost stable matchings in the roommates problem with bounded preference lists. Theor. Comput. Sci. 411, 1828–1841 (2010)zbMATHCrossRefGoogle Scholar
  5. 5.
    Cseh, Á., Irving, R.W., Manlove, D.F.: The stable roommates problem with short lists. CoRR abs/1605.04609 (2016). http://arxiv.org/abs/1605.04609
  6. 6.
    Feder, T.: A new fixed point approach for stable networks and stable marriages. J. Comput. Syst. Sci. 45, 233–284 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Feder, T.: Network flow and 2-satisfiability. Algorithmica 11, 291–319 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Monthly 69, 9–15 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Gusfield, D.: Three fast algorithms for four problems in stable marriage. SIAM J. Comput. 16(1), 111–128 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Gusfield, D., Irving, R.W.: The Stable Marriage Problem: Structure and Algorithms. MIT Press, Cambridge (1989)zbMATHGoogle Scholar
  11. 11.
    Gusfield, D., Pitt, L.: A bounded approximation for the minimum cost 2-SAT problem. Algorithmica 8, 103–117 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Hamada, K., Iwama, K., Miyazaki, S.: An improved approximation lower bound for finding almost stable maximum matchings. Inf. Process. Lett. 109, 1036–1040 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Irving, R.W.: An efficient algorithm for the "stable roommates" problem. J. Algorithms 6, 577–595 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Irving, R.W., Leather, P., Gusfield, D.: An efficient algorithm for the “optimal" stable marriage. J. ACM 34, 532–543 (1987)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Irving, R.W., Manlove, D.F.: The stable roommates problem with ties. J. Algorithms 43, 85–105 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Kujansuu, E., Lindberg, T., Mäkinen, E.: The stable roommates problem and chess tournament pairings. Divulgaciones Matemáticas 7, 19–28 (1999)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2-\(\varepsilon \). J. Comput. Syst. Sci. 74, 335–349 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Manlove, D.F.: Algorithmics of Matching Under Preferences. World Scientific, Singapore (2013)zbMATHCrossRefGoogle Scholar
  19. 19.
    Manlove, D.F., Irving, R.W., Iwama, K., Miyazaki, S., Morita, Y.: Hard variants of stable marriage. Theor. Comput. Sci. 276, 261–279 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Ronn, E.: NP-complete stable matching problems. J. Algorithms 11, 285–304 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Tan, J.J.M.: A necessary and sufficient condition for the existence of a complete stable matching. J. Algorithms 12, 154–178 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Teo, C.-P., Sethuraman, J.: LP based approach to optimal stable matchings. In: Proceedings of SODA 1997, pp. 710–719. ACM-SIAM (1997)Google Scholar
  23. 23.
    Teo, C.-P., Sethuraman, J.: The geometry of fractional stable matchings and its applications. Math. Oper. Res. 23, 874–891 (1998)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Ágnes Cseh
    • 1
  • Robert W. Irving
    • 2
  • David F. Manlove
    • 2
  1. 1.School of Computer ScienceReykjavik UniversityReykjavíkIceland
  2. 2.School of Computing ScienceUniversity of GlasgowGlasgowScotland, UK

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