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Sharing Supermodular Costs

Author

Listed:
  • Andreas S. Schulz

    (Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

  • Nelson A. Uhan

    (School of Industrial Engineering, Purdue University, West Lafayette, Indiana 47907)

Abstract

We study cooperative games with supermodular costs. We show that supermodular costs arise in a variety of situations; in particular, we show that the problem of minimizing a linear function over a supermodular polyhedron---a problem that often arises in combinatorial optimization---has supermodular optimal costs. In addition, we examine the computational complexity of the least core and least core value of supermodular cost cooperative games. We show that the problem of computing the least core value of these games is strongly NP-hard and, in fact, is inapproximable within a factor strictly less than 17/16 unless P = NP. For a particular class of supermodular cost cooperative games that arises from a scheduling problem, we show that the Shapley value---which, in this case, is computable in polynomial time---is in the least core, while computing the least core value is NP-hard.

Suggested Citation

  • Andreas S. Schulz & Nelson A. Uhan, 2010. "Sharing Supermodular Costs," Operations Research, INFORMS, vol. 58(4-part-2), pages 1051-1056, August.
    • Handle: RePEc:inm:oropre:v:58:y:2010:i:4-part-2:p:1051-1056
    DOI: 10.1287/opre.1100.0841
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    References listed on IDEAS

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    Cited by:

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    5. Chessa, Michela & Hanaki, Nobuyuki & Lardon, Aymeric & Yamada, Takashi, 2023. "An experiment on the Nash program: A comparison of two strategic mechanisms implementing the Shapley value," Games and Economic Behavior, Elsevier, vol. 141(C), pages 88-104.
    6. Simai He & Jiawei Zhang & Shuzhong Zhang, 2012. "Polymatroid Optimization, Submodularity, and Joint Replenishment Games," Operations Research, INFORMS, vol. 60(1), pages 128-137, February.
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    9. Lindong Liu & Xiangtong Qi & Zhou Xu, 2016. "Computing Near-Optimal Stable Cost Allocations for Cooperative Games by Lagrangian Relaxation," INFORMS Journal on Computing, INFORMS, vol. 28(4), pages 687-702, November.

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