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

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.

A new stable and more responsive cost sharing solution for minimum cost spanning tree problems

“…Bergantinos and Vidal-Puga (2007) show that CS is satisfied by the folk solution but not by the Kar solution. Lemma 5 shows that y cc satisfies CS.…”

“…We focus on the approach of Bergantinos and Vidal-Puga (2007), which uses the Shapley value, thus allowing for a clear comparison with the Kar solution.…”

“…We say that G is a chordal graph if, for all cycles of four or more nodes, there exists a chord, that is an edge connecting two non-adjacent nodes in the cycle. Ando (2010) shows that computing C and its Shapley value can be done in polynomial time if G(c, α) is a chordal graph for all α ∈ R + . Since c * is such that G(c * , α) is cycle-complete, it is clearly also a chordal graph.…”

“…Bird (1976) introduced the concept of irreducible cost matrix, on which is built the most famous cost allocation for mcst problems. First suggested by Feltkamp et al (1994) and rediscovered independently by Bergantinos and Vidal-Puga (2007), we follow Bogomolnaia and Moulin (2010) and call it the folk solution. We obtain the irreducible cost matrix as follows:…”

Monte Carlo Algorithm for Calculating the Shapley Values of Minimum Cost Spanning Tree Games

“…Let (N, f ) be a cooperative game. For a given permutation π of N , the marginal cost vector X(π) is defined as X(π) π(i) = f ({π (1)…”