Advertisement

Note on an Auction Procedure for a Matching Game in Polynomial Time

  • Winfried Hochstättler
  • Hui Jin
  • Robert Nickel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4041)

Abstract

We derive a polynomial time algorithm to compute a stable solution in a mixed matching market from an auction procedure as presented by Eriksson and Karlander [5]. As a special case we derive an \(\mathcal{O}(nm)\) algorithm for bipartite matching that does not seem to have appeared in the literature yet.

Keywords

Polynomial Time Algorithm Stable Match Assignment Game Preference List Stable Marriage 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows. Prentice-Hall, Englewood Cliffs (1993)Google Scholar
  2. 2.
    Demange, G., Gale, D., Sotomayor, M.: Multiitem auctions. Journal of Political Economy 94(4), 863–872 (1986)CrossRefGoogle Scholar
  3. 3.
    Deng, X., Papadimitriou, C.H.: On the complexity of cooparative game solution concepts. Mathematics of Operations Research 19, 257–266 (1994)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Deng, X., Ibaraki, T., Nagamochi, H.: Combinatorial optimization games. In: Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms, New Orleans, LA, pp. 720–729 (1997)Google Scholar
  5. 5.
    Eriksson, K., Karlander, J.: Stable matching in a common generalization of the marriage and assignment models. Discrete Mathematics 217(1-3), 135–156 (2000)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Faigle, U., Fekete, S.P., Hochstättler, W., Kern, W.: On the complexity of testing membership in the core of min cost spanning tree games. International Journal of Game Theory 26, 361–366 (1997)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Ford, L.R., Fulkerson, D.R.: A simple algorithm for finding maximal network flows and an application to the hitchcock problem. Canadian Journal of Mathematics 9, 210–218 (1957)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Frank, A.: On Kuhn’s Hungarian method – A tribute from Hungary. Technical report, Egerváry Research Group on Combinatorial Optimization (October 2004)Google Scholar
  9. 9.
    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. American Mathematical Monthly 69, 9–15 (1962)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Galil, Z.: Efficient algorithms for finding maximum matchings in graphs. ACM Computing Surveys 18(1), 23–38 (1986)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gusfield, D., Irving, R.W.: The stable marriage problem: Structure and algorithms. MIT Press, Cambridge (1989)MATHGoogle Scholar
  12. 12.
    Hochstättler, W., Jin, H., Nickel, R.: The hungarian method in a mixed matching market. Technical report, FernUniversität in Hagen, Germany (October 2005)Google Scholar
  13. 13.
    Knuth, D.E.: Stable marriage and its relation to other combinatorial problems. In: CRM Proceedings and Lecture Notes, vol. 10. American Mathematical Society (1997)Google Scholar
  14. 14.
    Kuhn, H.W.: The Hungarian method for the assignment problem. Naval Research Logistics Quaterly 2, 83–97 (1955)CrossRefGoogle Scholar
  15. 15.
    Kuhn, H.W.: Variants of the Hungarian method for the assignment problem. Naval Research Logistics Quaterly 3, 253–258 (1956)CrossRefGoogle Scholar
  16. 16.
    Roth, A.E., Sotomayor, M.: Stable outcomes in discrete and continuous models of two-sided matching: A unified treatment. Revista de Econometria, The Brazilian Review of Econometrics 16(2) (November 1996)Google Scholar
  17. 17.
    Shapley, L.S., Shubik, M.: The assignment game I: The core. International Journal of Game Theory 1, 111–130 (1972)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Sotomayor, M.: Existence of stable outcomes and the lattice property for a unified matching market. Mathematical Social Sciences 39, 119–132 (2000)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Winfried Hochstättler
    • 1
  • Hui Jin
    • 2
  • Robert Nickel
    • 1
  1. 1.Department of MathematicsFernUniversität in HagenHagen
  2. 2.Department of MathematicsBrandenburg Technical University CottbusCottbus

Personalised recommendations