Frontiers of Mathematics in China

, Volume 5, Issue 1, pp 47–63 | Cite as

Algorithms for core stability, core largeness, exactness, and extendability of flow games

  • Qizhi Fang
  • Rudolf Fleischer
  • Jian Li
  • Xiaoxun Sun
Research Article


We study core stability and some related properties of flow games defined on simple networks (all edge capacities are equal) from an algorithmic point of view. We first present a sufficient and necessary condition that can be tested efficiently for a simple flow game to have a stable core. We also prove the equivalence of the properties of core largeness, extendability, and exactness of simple flow games and provide an equivalent graph theoretic characterization which allows us to decide these properties in polynomial time.


Flow network series-parallel graph imputation cooperative game 


91A12 91A46 05C57 


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  1. 1.
    Bietenhader T, Okamoto Y. Core stability of minimum coloring games. In: Proceedings of 30th International Workshop on Graph-Theoretic Concept in Computer Science. Lecture Notes in Computer Science, Vol 3353. 2004, 389–401Google Scholar
  2. 2.
    Biswas A K, Parthasarathy T, Potters J A M, Voorneveld M. Large cores and exactness. Game and Economic Behavior, 1999, 28: 1–12MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Deng X, Ibaraki T, Nagamochi H. Algorithmic aspects of the core of combinatorial optimization games. Mathematics of Operations Research, 1999, 24: 751–766MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Deng X, Fang Q, Sun X. Finding nucleolus of flow game. In: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’06). 2006, 124–131Google Scholar
  5. 5.
    Deng X, Papadimitriou C H. On the complexity of cooperative solution concepts. Mathematics of Operations Research, 1994, 19: 257–266MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Duffin R. Topology of series-parallel networks. Journal of Mathematical Analysis and Applications, 1965, 10: 303–318MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Even S, Tarjan R E. Network flow and testing graph connectivity. SIAM J Comput, 1975, 4: 507–518MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Fang Q, Fleischer R, Li J, Sun X. Algorithms for core stability, core largeness, exactness and extendability of flow games. In: Proceedings of the 13th Annual International Computing and Combinatorics Conference. 2007, 439–447Google Scholar
  9. 9.
    Fang Q, Zhu S, Cai M, Deng X. Membership for core of LP games and other games. In: Lecture Notes in Computer Science, Vol 2108. 2001, 247–256CrossRefMathSciNetGoogle Scholar
  10. 10.
    Ford L R, Fulkerson D R. Flows in Networks. Princeton: Princeton University Press, 1962MATHGoogle Scholar
  11. 11.
    van Gellekom J R G, Potters J A M, Reijnierse J H. Prosperity properties of TUgames. International Journal of Game Theory, 1999, 28: 211–227MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Jakoby A, Liskiewicz M, Reischuk R. Space efficient algorithms for directed seriesparallel graphs. Journal of Algorithms, 2006, 60: 85–114MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kalai E, Zemel E. Totally balanced games and games of flow. Mathematics of Operations Research, 1982, 7: 476–478MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kalai E, Zemel E. Generalized network problems yielding totally balanced games. Operations Research, 1982, 30: 998–1008MATHCrossRefGoogle Scholar
  15. 15.
    Lucas W F. A game with no solution. Bulletin of the American Mathematical Society, 1968, 74: 237–239MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    von Neumann J, Morgenstern O. Theory of Games and Economic Behaviour. Princeton: Princeton University Press, 1944Google Scholar
  17. 17.
    Schrijver A. Combinatorial Optimization: Polyhedra and Efficiency. Berlin: Springer-Verlag, 2003MATHGoogle Scholar
  18. 18.
    Shapley L S. Cores and convex games. International Journal of Game Theory, 1971, 1: 11–26MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Sharkey W W. Cooperative games with large cores. International Journal of Game Theory, 1982, 11: 175–182MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Solymosi T, Raghavan T E S. Assignment games with stable cores. International Journal of Game Theory, 2001, 30: 177–185MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Sun X, Fang Q. Core stability of flow games. In: Proceedings of the 2005 China-Japan Conference on Discrete Geometry, Combinatorics and Graph Theory. 2007, 189–199Google Scholar
  22. 22.
    Tholey T. Solving the 2-disjoint paths problem in nearly linear time. Theory of Computing Systems, 2006, 39: 51–78MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Valdes J, Tarjan R E, Lawler E L. The recognition of series-parallel digraphs. In: Proceedings of the 11th Annual ACM Symposium on Theory of Computing (STOC’79). 1979, 1–12Google Scholar

Copyright information

© Higher Education Press and Springer Berlin Heidelberg 2009

Authors and Affiliations

  • Qizhi Fang
    • 1
  • Rudolf Fleischer
    • 2
  • Jian Li
    • 3
  • Xiaoxun Sun
    • 4
  1. 1.Department of MathematicsOcean University of ChinaQingdaoChina
  2. 2.Department of Computer Science and EngineeringFudan UniversityShanghaiChina
  3. 3.Department of Computer ScienceUniversity of MarylandCollege ParkUSA
  4. 4.Department of Mathematics and ComputingUniversity of Southern QueenslandToowoombaAustralia

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