The Label-Constrained Minimum Spanning Tree Problem

Given a positive integer K and a connected, undirected graph G whose edges are labeled (or colored) and have weights, the label-constrained minimum spanning tree (LCMST) problem seeks a minimum weight spanning tree with at most K distinct labels (or colors). In this paper, we prove that the LCMST problem is NP-complete. Next, we introduce two local search methods to solve the problem. Then, we present a genetic algorithm which gets comparable results, but is much faster. In addition, we present two mixed integer programming formulations for the LCMST problem. We compare these on some small problem instances. Finally, we introduce a dual problem.