We consider the following problem. Given a graph G = (V, E), a partition of E into k color classes, E = ∪k i=1 Ei, and a cost function for each class fi: 2E i ↦→ R+, find a spanning tree T = (V, F) whose total cost is minimal, where the cost ofi(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.
|Number of pages
|Australasian Journal of Combinatorics
|Published - Jun 2019
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ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics