Network design through forests with degree- and role-constrained minimum spanning trees

Finding the degree-constrained minimum spanning tree (DCMST) of a graph is a widely studied NP-hard problem. One of its most important applications is network design. Here we deal with a new variant of the DCMST problem, which consists of finding not only the degree- but also the role-constrained minimum spanning tree (DRCMST), i.e., we add constraints to restrict the role of the nodes in the tree to root, intermediate or leaf node. Furthermore, we do not limit the number of root nodes to one, thereby, generally, building a forest of DRCMSTs. The modeling of network design problems can benefit from the possibility of generating more than one tree and determining the role of the nodes in the network. We propose a novel permutation-based representation to encode these forests. In this new representation, one permutation simultaneously encodes all the trees to be built. We simulate a wide variety of DRCMST problem instances which we optimize using different evolutionary computation algorithms encoding individuals of the population using the proposed representation. To illustrate the applicability of our approach, we formulate the trans-European transport network as a DRCMST problem. In this network design, we simultaneously optimize nine transport corridors and show that it is straightforward using the proposed representation to add constraints depending on the specific characteristics of the network.