Parallel heuristics for the bounded diameter minimum spanning tree problem

Abstract

Given a connected, undirected graph G = (V, E) on n = |V| vertices, an integer diameter bound D ≥ 2 and non-zero edge weights associated with each edge e ∈ 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) = Σ ∀ e ∈ T w(e) and tree diameter no greater than D. The problem of computing BDMSTs is known to be NP-Hard for 4 ≤ D < n−1, and hard to approximate. Well known greedy and randomized greedy heuristic strategies are extant in the literature which build low cost diameter-constrained spanning trees in O(n3) time. The greedy heuristics perform better on graphs with random edge weights, whereas the randomized greedy heuristics produce lower cost BDSTs on Euclidean graphs. Another recent heuristic uses a “less greedy” approach and performs competitively vis-a-vis the other heuristics. This paper presents parallel versions of some of these heuristics (the Center-based Tree Construction (CBTC), Randomized Tree Construction (RTC) and Center-based Least Sum-of-costs (CBLSoC) heuristics) implemented using OpenMP. The reason for choosing these heuristics for parallelization is that they have been shown to perform well over a wide variety of problem instances, whereas other extant heuristics in the literature are known to give perform well only on specific types of graphs. The performance of all the heuristics is compared on a wide range of standard benchmark instances comprising of completely connected Euclidean and random graphs.