Fast Heuristics for Large Instances of the Euclidean Bounded Diameter Minimum Spanning Tree Problem (Informatica, 2015)
Given a connected, undirected graph G = (V, E) on n = jV j vertices, an integer bound D>=2 and non-zeroedge weights associated with each edge e 2 E, a bounded diameter minimum spanning tree (BDMST) on Gis 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 nogreater than D. The Euclidean BDMST Problem aims to find the minimum cost BDMST on graphs whosevertices are points in Euclidean space and whose edge weights are the Euclidean distances between thecorresponding 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. Heuristicsare extant in the literature which build low cost, diameter-constrained spanning trees in O(n3) time. Thispaper presents some fast and effective heuristic strategies for the Euclidean BDMST Problem and comparestheir performance with that of the best known existing heuristics. Two of the proposed heuristics run inO(n2pn) time and another faster heuristic runs in O(n2), thereby allowing them to quickly build low costBDSTs on larger sized problems than have been attempted hitherto. The proposed heuristics are shown toperform better over a wide range of benchmark instances used in the literature for the Euclidean BDMSTProblem. Further, a new test suite of much larger problem sizes than attempted hitherto in the literature isdesigned and results presented.