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Computation of the Shapley value of minimum cost spanning tree games: #P-hardness and polynomial cases

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Abstract

We show that computing the Shapley value of minimum cost spanning tree games is #P-hard even if the cost functions are restricted to be {0,1}-valued. The proof is by a reduction from counting the number of minimum 2-terminal vertex cuts of an undirected graph, which is #P-complete. We also investigate minimum cost spanning tree games whose Shapley values can be computed in polynomial time. We show that if the cost function of the given network is a subtree distance, which is a generalization of a tree metric, then the Shapley value of the associated minimum cost spanning tree game can be computed in O(n 4) time, where n is the number of players.

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Authors and Affiliations

  1. Faculty of Engineering, Shizuoka University, Hamamatsu, 432-8561, Japan

    Kazutoshi Ando

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  1. Kazutoshi Ando
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Correspondence to Kazutoshi Ando.

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This work is supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C) (No. 23510066).

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Ando, K. Computation of the Shapley value of minimum cost spanning tree games: #P-hardness and polynomial cases. Japan J. Indust. Appl. Math. 29, 385–400 (2012). https://doi.org/10.1007/s13160-012-0078-9

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  • DOI: https://doi.org/10.1007/s13160-012-0078-9

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