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

Abstract

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.