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Stable Marriage and Roommates Problems with Restricted Edges: Complexity and Approximability

  • Ágnes Cseh
  • David F. Manlove
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9347)

Abstract

In the stable marriage and roommates problems, a set of agents is given, each of them having a strictly ordered preference list over some or all of the other agents. A matching is a set of disjoint pairs of mutually acceptable agents. If any two agents mutually prefer each other to their partner, then they block the matching, otherwise, the matching is said to be stable. We investigate the complexity of finding a solution satisfying additional constraints on restricted pairs of agents. Restricted pairs can be either forced or forbidden. A stable solution must contain all of the forced pairs, while it must contain none of the forbidden pairs.

Dias et al. [5] gave a polynomial-time algorithm to decide whether such a solution exists in the presence of restricted edges. If the answer is no, one might look for a solution close to optimal. Since optimality in this context means that the matching is stable and satisfies all constraints on restricted pairs, there are two ways of relaxing the constraints by permitting a solution to: (1) be blocked by as few as possible pairs, or (2) violate as few as possible constraints on restricted pairs.

Our main theorems prove that for the (bipartite) stable marriage problem, case (1) leads to \(\mathsf{NP}\)-hardness and inapproximability results, whilst case (2) can be solved in polynomial time. For non-bipartite stable roommates instances, case (2) yields an \(\mathsf{NP}\)-hard but (under some cardinality assumptions) 2-approximable problem. In the case of NP-hard problems, we also discuss polynomially solvable special cases, arising from restrictions on the lengths of the preference lists, or upper bounds on the numbers of restricted pairs.

Notes

Acknowledgements

We would like to thank the anonymous reviewers for their valuable comments, which have helped to improve the presentation of this paper.

References

  1. 1.
    Ageev, A.A., Kononov, A.V.: Approximation algorithms for scheduling problems with exact delays. In: Erlebach, T., Kaklamanis, C. (eds.) WAOA 2006. LNCS, vol. 4368, pp. 1–14. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  2. 2.
    Biró, P., Manlove, D., McDermid, E.: “Almost stable” matchings in the roommates problem with bounded preference lists. Theor. Comput. Sci. 432, 10–20 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Biró, P., Manlove, D., Mittal, S.: Size versus stability in the marriage problem. Theor. Comput. Sci. 411, 1828–1841 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cseh, A., Manlove, D.: Stable marriage and roommates problems with restricted edges: complexity and approximability. Technical Report 1412.0271, Computing Research Repository, Cornell University Library, 2015. http://arxiv.org/abs/1412.0271
  5. 5.
    Dias, V., da Fonseca, G., de Figueiredo, C., Szwarcfiter, J.: The stable marriage problem with restricted pairs. Theor. Comput. Sci. 306(1–3), 391–405 (2003)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Feder, T.: A new fixed point approach for stable networks and stable marriages. J. Comput. Syst. Sci. 45, 233–284 (1992)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Feder, T.: Network flow and 2-satisfiability. Algorithmica 11(3), 291–319 (1994)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fleiner, T., Irving, R., Manlove, D.: Efficient algorithms for generalised stable marriage and roommates problems. Theor. Comput. Sci. 381(1–3), 162–176 (2007)CrossRefMATHGoogle Scholar
  9. 9.
    Gai, A.-T., Lebedev, D., Mathieu, F., de Montgolfier, F., Reynier, J., Viennot, L.: Acyclic preference systems in P2P networks. In: Kermarrec, A.-M., Bougé, L., Priol, T. (eds.) Euro-Par 2007. LNCS, vol. 4641, pp. 825–834. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  10. 10.
    Gale, D., Shapley, L.: College admissions and the stability of marriage. Am. Math. Monthly 69, 9–15 (1962)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gale, D., Sotomayor, M.: Some remarks on the stable matching problem. Discrete Appl. Math. 11, 223–232 (1985)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gusfield, D., Irving, R.: The Stable Marriage Problem: Structure and Algorithms. MIT Press, Cambridge (1989)MATHGoogle Scholar
  13. 13.
    Hamada, K., Iwama, K., Miyazaki, S.: An improved approximation lower bound for finding almost stable maximum matchings. Inf. Process. Lett. 109(18), 1036–1040 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Irving, R.: An efficient algorithm for the “stable roommates” problem. J. Algorithms 6, 577–595 (1985)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Irving, R., Leather, P., Gusfield, D.: An efficient algorithm for the “optimal” stable marriage. J. ACM 34(3), 532–543 (1987)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Irving, R., Manlove, D.: The stable roommates problem with ties. J. Algorithms 43, 85–105 (2002)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Khot, S., Regev, O.: Vertex cover might be hard to approximate to within \(2-\varepsilon \). J. Comput. Syst. Sci. 74(3), 335–349 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Khuller, S., Mitchell, S., Vazirani, V.: On-line algorithms for weighted bipartite matching and stable marriages. Theor. Comput. Sci. 127, 255–267 (1994)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Király, T., Pap, J.: Total dual integrality of Rothblum’s description of the stable-marriage polyhedron. Math. Oper. Res. 33(2), 283–290 (2008)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Knuth, D.: Mariages Stables. (Les Presses de L’Université de Montréal, 1976). English translation in Stable Marriage and its Relation to Other Combinatorial Problems. CRM Proceedings and Lecture Notes, vol. 10. American Mathematical Society (1997)Google Scholar
  21. 21.
    O’Malley, G.: Algorithmic Aspects of Stable Matching Problems. Ph.D thesis, University of Glasgow, Department of Computing Science (2007)Google Scholar
  22. 22.
    Roth, A.: The evolution of the labor market for medical interns and residents: a case study in game theory. J. Political Econ. 92(6), 991–1016 (1984)CrossRefGoogle Scholar
  23. 23.
    Roth, A.: Deferred acceptance algorithms: history, theory, practice, and open questions. Int. J. Game Theor. 36(3–4), 537–569 (2008)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Roth, A., Sotomayor, M.: Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis. Econometric Society Monographs, vol. 18. Cambridge University Press, Cambridge (1990) CrossRefMATHGoogle Scholar
  25. 25.
    Teo, C.-P., Sethuraman, J.: The geometry of fractional stable matchings and its applications. Math. Oper. Res. 23(4), 874–891 (1998)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institute for MathematicsTU BerlinBerlinGermany
  2. 2.School of Computing ScienceUniversity of GlasgowGlasgowUK

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