Research Article Degree-Constrained k-Minimum Spanning Tree Problem Pablo Adasme 1 and Ali Dehghan Firoozabadi 2 1 Department of Electrical Engineering, Universidad de Santiago de Chile, Avenida Ecuador 3519, Santiago, Chile 2 Department of Electricity, Universidad Tecnol´ ogica Metropolitana, Av. Jose Pedro Alessandri 1242, 7800002 Santiago, Chile Correspondence should be addressed to Pablo Adasme; pablo.adasme@usach.cl Received 15 June 2020; Revised 25 August 2020; Accepted 10 October 2020; Published 9 November 2020 Academic Editor: Qingling Wang Copyright © 2020 Pablo Adasme and Ali Dehghan Firoozabadi. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let G(V, E) 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 (DCkMST) problem which consists of finding a minimum cost subtree of G formed with at least k vertices of V wherethedegreeofeachvertexislessthanorequaltoanintegervalue d ≤ k − 2. In particular, in this paper, we consider degree values of d ∈ 2, 3 { }. Notice that DCkMSTgeneralizes 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 DCkMST can be adapted or directly utilized include backbone network structures in telecom- munications, facility location, and transportation networks, to name a few. It is easy to see from the literature that the DCkMST problemhasnotbeenstudiedindepthsofar.us,ourmaincontributionsinthispapercanbehighlightedasfollows.Wepropose three mixed-integer linear programming (MILP) models for the DCkMST problem and derive for each one an equivalent counterpart by using the handshaking lemma. en, we further propose ant colony optimization (ACO) and variable neigh- borhood 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 generatedoneswithuniformandEuclideandistancecostswithupto400nodes.Ournumericalresultsindicatethattheproposed 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. 1.Introduction Intelligent communication systems will be mandatorily required in the next decades to provide low-cost connec- tivity within many application domains involving network structuresintheformofring,tree,andstartopologies[1–3], for instance, when designing network structures in tele- communications, facility location, electrical power systems, water and transportation networks, and for networks to be constructed under the Internet of ings (IoT) paradigm [2–10]. A particular example related with wireless sensor networksisduetodensitycontrolmethods[1,11,12].ese methods allow reducing energy consumption in sensor networks by means of activation or deactivation mechanisms which mainly consist of putting into sleep mode some of the nodes of the network while ensuring sensing operations, communication, and connectivity [1–3, 12–16]. Let G(V, E) represent a simple undirected complete graph with set of nodes V � 1, ... ,n { } and set of edges E � 1, ... ,m { }. In this paper, we consider the degree-con- strained k-minimum spanning tree problem (DCkMST), which consists of finding a minimum cost subtree of G formedwithatleast k ≤ n vertices of V whereeachvertexhas a degree lower than or equal to d ∈ 2, ... ,k − 2 { }. Notice that, for d � 2, DCkMST reduces to find a minimum cost Hamiltonian path with cardinality k.eDCkMSTproblem generalizes two classical combinatorial optimization Hindawi Complexity Volume 2020, Article ID 7628105, 25 pages https://doi.org/10.1155/2020/7628105