A fix-and-optimize matheuristic for the k-labelled spanning forest problem

In this paper, we study the k-labeled spanning forest problem (kLSF). The input for this problem is an undirected graph with labeled edges and a positive integer k. The goal is to find a spanning forest of the graph with at most $k$ different labels associated with the edges, minimizing the number of components. kLSF finds practical applications in different scenarios related to networks design and telecommunications. Solving it may help to reduce the negative impact of electromagnetic fields exposure on the population health or to increase profits of internet management companies, among others. The interest in kLSF is not only practical but also theoretical since the problem generalizes the best-known NP-hard minimum labeling spanning tree problem (MLST). To approach kLSF, we propose a fix-and-optimize matheuristic that was tested over several instances, achieving high-quality solutions in reasonable computational time. When compared to the best-known algorithms in the literature, our matheuristic outperformed the other proposals in most cases, finding better solutions in less computational time for the most challenging instances.