A novel Hybrid Multi-objective Evolutionary Algorithm for the bi-Objective Minimum Diameter-Cost Spanning Tree (bi-MDCST) problem

Abstract

Given a connected, weighted, undirected graph G = (V, E), the bi-objective Minimum Diameter-Cost Spanning Tree (bi-MDCST) Problem aims to find spanning trees on G with minimal total edge weight as well as minimal diameter. The problem is known to be NP-hard and finds application in domains ranging from transportation to network design. An exact algorithm for the problem is given in the literature, but is computationally very expensive and hence applicable to only very small graphs; some fast heuristics, a multi-objective genetic algorithm (MOGA) and a fast non-dominated sorting genetic algorithm (NSGA2) have also been proposed in the literature for solving the problem and their performance studied on larger sized full graphs containing up to 250 vertices and 31,125 edges. This paper presents a novel Hybrid Multi-objective Evolutionary Algorithm (HMOEA) for the bi-MDCST problem which uses a representation that captures the genetic information of several spanning trees of varying diameters in a single chromosome, thereby enhancing the representational power of the population. The algorithm uses heuristic-based population seeding and combines linear time recombination with a fast mutation operator to more effectively balance the exploration and exploitation of the search space. The proposed HMOEA computes the optimal Pareto-front solutions in time that is several orders of magnitude lesser than that taken by the exact method, and is shown to obtain superior performance in terms of the convergence and distribution of solutions vis-à-vis the other two extant meta-heuristics on a wide range of benchmark graph instances. Results obtained by the HMOEA are also reported for larger sized Euclidean instances than attempted hitherto in the literature, containing up to 500 vertices and 124,750 edges, and for a benchmark suite of large fully connected random edge-weight graph instances introduced in this work for more effectively comparing the performance of algorithms for the bi-MDCST problem.