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Solutions for the Stable Roommates Problem with Payments

  • Péter Biró
  • Matthijs Bomhoff
  • Petr A. Golovach
  • Walter Kern
  • Daniël Paulusma
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7551)

Abstract

The stable roommates problem with payments has as input a graph G = (V,E) with an edge weighting w: E → ℝ +  and the problem is to find a stable solution. A solution is a matching M with a vector \(p\in{\mathbb R}^V_+\) that satisfies p u  + p v  = w(uv) for all uv ∈ M and p u  = 0 for all u unmatched in M. A solution is stable if it prevents blocking pairs, i.e., pairs of adjacent vertices u and v with p u  + p v  < w(uv). By pinpointing a relationship to the accessibility of the coalition structure core of matching games, we give a simple constructive proof for showing that every yes-instance of the stable roommates problem with payments allows a path of linear length that starts in an arbitrary unstable solution and that ends in a stable solution. This result generalizes a result of Chen, Fujishige and Yang for bipartite instances to general instances. We also show that the problems Blocking Pairs and Blocking Value, which are to find a solution with a minimum number of blocking pairs or a minimum total blocking value, respectively, are NP-hard. Finally, we prove that the variant of the first problem, in which the number of blocking pairs must be minimized with respect to some fixed matching, is NP-hard, whereas this variant of the second problem is polynomial-time solvable.

Keywords

Stable Solution Weighted Graph Adjacent Vertex Stable Match Assignment Game 
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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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.
    Béal, S., Rémila, E., Solal, P.: On the number of blocks required to access the coalition structure core. Working Paper, Munich Personal RePEc Archive, MPRA Paper No. 29755 (2011)Google Scholar
  3. 3.
    Biró, P., Kern, W., Paulusma, D.: Computing solutions for matching games. International Journal of Game Theory 41, 75–90 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Chen, B., Fujishige, S., Yang, Z.: Decentralized Market Processes to Stable Job Matchings with Competitive Salaries. Working Paper, Kyoto University, RIMS-1715 (2011)Google Scholar
  5. 5.
    Diamantoudi, E., Miyagawa, E., Xue, L.: Random paths to stability in the roommates problem. Games and Economic Behavior 48, 18–28 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Eriksson, K., Karlander, J.: Stable outcomes of the roommate game with transferable utility. International Journal of Game Theory 29, 555–569 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. American Mathematical Monthly 69, 9–15 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  9. 9.
    Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete graph problems. Theoret. Comput. Sci. 1, 237–267 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Khachiyan, L.G.: A polynomial algorithm in linear programming. Soviet Mathematics Doklady 20, 191–194 (1979)zbMATHGoogle Scholar
  11. 11.
    Kóczy, L.Á., Lauwers, L.: The coalition structure core is accessible. Games and Economic Behavior 48, 86–93 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Knuth, D.E.: Mariages stable et leurs relations avec d’autres problèmes combinatoires. Les Presses de l’Université de Montréal, Montréal (1976)Google Scholar
  13. 13.
    Roth, A.E., Vande Vate, J.H.: Random paths to stability in two-sided matching. Econometrica 58, 1475–1480 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Shapley, L.S., Shubik, M.: The assignment game I: the core. International Journal of Game Theory 1, 111–130 (1972)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Yang, Y.-Y.: Accessible outcomes versus absorbing outcomes. Mathematical Social Sciences 62, 65–70 (2011)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Péter Biró
    • 1
  • Matthijs Bomhoff
    • 2
  • Petr A. Golovach
    • 3
  • Walter Kern
    • 2
  • Daniël Paulusma
    • 3
  1. 1.Institute of EconomicsHungarian Academy of SciencesBudapestHungary
  2. 2.Faculty of Electrical Engineering, Mathematics and Computer ScienceUniversity of TwenteEnschedeThe Netherlands
  3. 3.School of Engineering and Computing SciencesDurham University, Science LaboratoriesDurhamUK

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