Degree-Constrained k -Minimum Spanning Tree Problem

Abstract

Let $G\left(V,E\right)$ be a simple undirected complete graph with vertex and edge sets $V$ and $E$ , respectively. In this paper, we consider the degree-constrained $k$ -minimum spanning tree (DC $k$ MST) problem which consists of finding a minimum cost subtree of $G$ formed with at least $k$ vertices of $V$ where the degree of each vertex is less than or equal to an integer value $d\le k-2$ . In particular, in this paper, we consider degree values of $d\in \left\{\mathrm{2,3}\right\}$ . Notice that DC $k$ MST generalizes both the classical degree-constrained and $k$ -minimum spanning tree problems simultaneously. In particular, when $d=2$ , it reduces to a $k$ -Hamiltonian path problem. Application domains where DC $k$ MST can be adapted or directly utilized include backbone network structures in telecommunications, facility location, and transportation networks, to name a few. It is easy to see from the literature that the DC $k$ MST problem has not been studied in depth so far. Thus, our main contributions in this paper can be highlighted as follows. We propose three mixed-integer linear programming (MILP) models for the DC $k$ MST problem and derive for each one an equivalent counterpart by using the handshaking lemma. Then, we further propose ant colony optimization (ACO) and variable neighborhood search (VNS) algorithms. Each proposed ACO and VNS method is also compared with another variant of it which is obtained while embedding a Q-learning strategy. We also propose a pure Q-learning algorithm that is competitive with the ACO ones. Finally, we conduct substantial numerical experiments using benchmark input graph instances from TSPLIB and randomly generated ones with uniform and Euclidean distance costs with up to 400 nodes. Our numerical results indicate that the proposed models and algorithms allow obtaining optimal and near-optimal solutions, respectively. Moreover, we report better solutions than CPLEX for the large-size instances. Ultimately, the empirical evidence shows that the proposed Q-learning strategies can bring considerable improvements.