## Abstract

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.

Original language | English |
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Pages (from-to) | 33-45 |

Number of pages | 13 |

Journal | Australasian Journal of Combinatorics |

Volume | 74 |

Issue number | 1 |

State | Published - Jun 2019 |

### Bibliographical note

Publisher Copyright:© The author(s).

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics