An effective genetic algorithm approach to the quadratic minimum spanning tree problem

Abstract

Scope and Purpose—The minimum spanning tree (MST) problem is one of the traditional optimization problems. Its linear objective function makes it easy to deal with. However, if we consider the presence of the interaction effects of the cost between pairs of edges, it results in a new minimum spanning tree problem with a quadratic (rather linear) objective function, which is denoted as the quadratic minimum spanning tree (q-MST) problem. This q-MST problem is of high importance in network optimization design such as oil transportation, communication, etc. Because the problem is NP-hard and no effective algorithms exist, we have developed a new approach by using a genetic algorithm (GA) to deal with it. In this paper we present a new approach to solve the q-MST problem by using a genetic algorithm. A skillful encoding for trees, denoted by Prüfer number, is adopted for GA operation. On comparing with the existing heuristic algorithms by 17 randomly generated numerical examples from 6-vertex graph to 50-vertex graph, the new GA approach shows its high effectiveness in solving the q-MST problem and real value in the practical network optimization.