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

摘要

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.