Degree-Constrained  k -Minimum Spanning Tree Problem

Pablo Adasme,Ali Dehghan Firoozabadi

doi:10.1155/2020/7628105

Degree-Constrained k -Minimum Spanning Tree Problem

Pablo Adasme, Ali Dehghan Firoozabadi

Open Access

https://doi.org/10.1155/2020/7628105

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Abstract

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 (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 ≤ k − 2 . In particular, in this paper, we consider degree values of d ∈ 2,3 . 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.

Highlights

We propose three mixed-integer linear programming (MILP) models for the degree-constrained k-minimum spanning tree (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 neighborhood search (VNS) algorithms
Intelligent communication systems will be mandatorily required in the decades to provide low-cost connectivity within many application domains involving network structures in the form of ring, tree, and star topologies [1,2,3], for instance, when designing network structures in telecommunications, facility location, electrical power systems, water and transportation networks, and for networks to be constructed under the Internet of ings (IoT) paradigm [2,3,4,5,6,7,8,9,10]
Columns 14 and 15 present gaps that we compute by [(Opt − LP)/Opt]∗100 where Opt and LP refer to the optimal solution found with the MILP and LP models, respectively

Summary

Introduction

Intelligent communication systems will be mandatorily required in the decades to provide low-cost connectivity within many application domains involving network structures in the form of ring, tree, and star topologies [1,2,3], for instance, when designing network structures in telecommunications, facility location, electrical power systems, water and transportation networks, and for networks to be constructed under the Internet of ings (IoT) paradigm [2,3,4,5,6,7,8,9,10]. For the VNS algorithm, we further introduce an embedded Q-learning strategy that allows performing a random local search based on the experience of previous solutions found [31]. The authors propose a reactive search VNS method that allows learning which is the order in which different local search heuristics must be applied in order to obtain better solutions based on the experience of previous trials. E embedded Q-learning strategy in our VNS approach is different as we use it as a learning mechanism in order to perform a random local search It does not require the use of specialized local search methods, and it can be extended and used straightforwardly for any other combinatorial optimization problems.

Related Work

Mathematical Formulations

Numerical Experiments

Conclusions