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On computational complexity of membership test in flow games and linear production games


  • Shanfeng Zhu

    (Department of Computer Science, City University of Hong Kong, Hong Kong, P. R. China)

  • Xiaotie Deng

    (Department of Computer Science, City University of Hong Kong, Hong Kong, P. R. China)

  • Maocheng Cai

    (Institute of Systems Science, Chinese Academy of Sciences, Beijing 100080, P. R. China)

  • Qizhi Fang

    (Department of Mathematics, Ocean University of Qingdao, Qingdao 266003, P. R. China)


Let \equiv(N,v) be a cooperative game with the player set N and characteristic function v: 2N ->R. An imputation of the game is in the core if no subset of players could gain advantage by splitting from the grand coalition of all players. It is well known that, for the flow game (and equivalently, for the linear production game), the core is always non-empty and a solution in the core can be found in polynomial time. In this paper, we show that, given an imputation x, it is NP-complete to decide x is not a member of the core, for the flow game. And because of the specific reduction we constructed, the result also holds for the linear production game.

Suggested Citation

  • Shanfeng Zhu & Xiaotie Deng & Maocheng Cai & Qizhi Fang, 2002. "On computational complexity of membership test in flow games and linear production games," International Journal of Game Theory, Springer;Game Theory Society, vol. 31(1), pages 39-45.
  • Handle: RePEc:spr:jogath:v:31:y:2002:i:1:p:39-45
    Note: Received: October 2000/Final version: March 2002

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    Cited by:

    1. Demuynck, Thomas, 2011. "The computational complexity of rationalizing boundedly rational choice behavior," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 425-433.
    2. Kumabe, Masahiro & Mihara, H. Reiju, 2008. "Computability of simple games: A characterization and application to the core," Journal of Mathematical Economics, Elsevier, vol. 44(3-4), pages 348-366, February.
    3. Kumabe, Masahiro & Mihara, H. Reiju, 2011. "Computability of simple games: A complete investigation of the sixty-four possibilities," Journal of Mathematical Economics, Elsevier, vol. 47(2), pages 150-158, March.
    4. Potters, Jos & Reijnierse, Hans & Biswas, Amit, 2006. "The nucleolus of balanced simple flow networks," Games and Economic Behavior, Elsevier, vol. 54(1), pages 205-225, January.
    5. Drechsel, J. & Kimms, A., 2010. "Computing core allocations in cooperative games with an application to cooperative procurement," International Journal of Production Economics, Elsevier, vol. 128(1), pages 310-321, November.
    6. Mohammaditabar, Davood & Ghodsypour, Seyed Hassan & Hafezalkotob, Ashkan, 2016. "A game theoretic analysis in capacity-constrained supplier-selection and cooperation by considering the total supply chain inventory costs," International Journal of Production Economics, Elsevier, vol. 181(PA), pages 87-97.


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