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Computing solutions for matching games

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Author Info

  • Péter Biró

    ()

  • Walter Kern

    ()

  • Daniël Paulusma

    ()

Abstract

A matching game is a cooperative game (N; v) defined on a graph G = (N;E) with an edge weighting w : E ! R+. The player set is N and the value of a coalition S  N is de ned as the maximum weight of a matching in the subgraph induced by S. First we present an O(nm+n2 log n) algorithm that tests if the core of a matching game defined on a weighted graph with n vertices and m edges is nonempty and that computes a core member if the core is nonempty. This algorithm improves previous work based on the ellipsoid method and can also be used to compute stable solutions for instances of the stable roommates problem with payments. Second we show that the nucleolus of an n-player matching game with a nonempty core can be computed in O(n4) time. This generalizes the corresponding result of Solymosi and Raghavan for assignment games. Third we prove that is NP-hard to determine an imputation with minimum number of blocking pairs, even for matching games with unit edge weights, whereas the problem of determining an imputation with minimum total blocking value is shown to be polynomial-time solvable for general matching games.

(This abstract was borrowed from another version of this item.)

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File URL: http://hdl.handle.net/10.1007/s00182-011-0273-y
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Bibliographic Info

Article provided by Springer in its journal International Journal of Game Theory.

Volume (Year): 41 (2012)
Issue (Month): 1 (February)
Pages: 75-90

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Handle: RePEc:spr:jogath:v:41:y:2012:i:1:p:75-90

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Related research

Keywords: Matching game; Nucleolus; Cooperative game theory;

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References

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  1. Johan Karlander & Kimmo Eriksson, 2001. "Stable outcomes of the roommate game with transferable utility," International Journal of Game Theory, Springer, vol. 29(4), pages 555-569.
  2. M. L. Balinski, 1965. "Integer Programming: Methods, Uses, Computations," Management Science, INFORMS, vol. 12(3), pages 253-313, November.
  3. Jeroen Kuipers & Ulrich Faigle & Walter Kern, 2001. "On the computation of the nucleolus of a cooperative game," International Journal of Game Theory, Springer, vol. 30(1), pages 79-98.
  4. Solymosi, Tamas & Raghavan, Tirukkannamangai E S, 1994. "An Algorithm for Finding the Nucleolus of Asignment Games," International Journal of Game Theory, Springer, vol. 23(2), pages 119-43.
  5. Bettina Klaus & Alexandru Nichifor, 2010. "Consistency in one-sided assignment problems," Social Choice and Welfare, Springer, vol. 35(3), pages 415-433, September.
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Cited by:
  1. P‚ter Bir¢ & Matthijs Bomhoff & Walter Kern & Petr A. Golovach & Dani‰l Paulusma, 2012. "Solutions for the Stable Roommates Problem with Payments," IEHAS Discussion Papers 1211, Institute of Economics, Hungarian Academy of Sciences.

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