On the Complexity of Testing Membership in the Core of min-Cost Spanning Tree Games

Let N = {1, ... ,n} be a finite set of players and K N the complete graph on the node set N w {0}. Assume that the edges of K N have nonnegative weights and associate with each coalition S _~ N of players as cost c(S) the weight of a minimal spanning tree on the node set S u {0}. Using transformation from EXACT COVER BY 3-SETS, we exhibit the tollowing problem to be NP-complete. Given the vector x~9t N with x(N) = c(N). decide whether there exists a coalition S such that x(S) > c(S).