Contractors’ minimum spanning tree

We consider the following problem. Given a graph G = (V, E), a partition of E into k color classes, E = ∪^k _i=1 E_i, and a cost function for each class f_i: 2^{E i} ↦→ R⁺, find a spanning tree T = (V, F) whose total cost is minimal, where the cost of_i(FT is defined as the sum of the costs of the color classes in T, namely_(formula)We show that the general problem is NP-hard, even when the cost functions depend only on the number of edges and are discrete and concave. We also provide a characterization of when a tree, with a prescribed number of edges from each color class, exists, as well as an efficient algorithm for finding such a tree. Finally, we prove that the polytope of feasible solutions for cardinality cost functions values is integral.