Fast Heuristics for Large Instances of the Euclidean Bounded Diameter Minimum Spanning Tree Problem

Given a connected, undirected graph G = (V, E) on n = jV j vertices, an integer bound D>=2 and non-zero edge weights associated with each edge e 2 E, a bounded diameter minimum spanning tree (BDMST) on G is defined as a spanning tree T E on G of minimum edge cost w(T) = P w(e), 8 e 2 T and tree diameter no greater than D. The Euclidean BDMST Problem aims to find the minimum cost BDMST on graphs whose vertices are points in Euclidean space and whose edge weights are the Euclidean distances between the corresponding vertices. The problem of computing BDMSTs is known to be NP-Hard for 4 D n -1, where D the diameter bound. Furthermore, the problem is known to be hard to approximate. Heuristics are extant in the literature which build low cost, diameter-constrained spanning trees in O(n3) time. This paper presents some fast and effective heuristic strategies for the Euclidean BDMST Problem and compares their performance with that of the best known existing heuristics. Two of the proposed heuristics run in O(n2pn) time and another faster heuristic runs in O(n2), thereby allowing them to quickly build low cost BDSTs on larger sized problems than have been attempted hitherto. The proposed heuristics are shown to perform better over a wide range of benchmark instances used in the literature for the Euclidean BDMST Problem. Further, a new test suite of much larger problem sizes than attempted hitherto in the literature is designed and results presented.