Novel Heuristics for the Euclidean Leaf-Constrained Minimum Spanning Tree Problem

Abstract

Given a graph G = (V, E) whose vertices are points in the two-dimensional Euclidean space and edge-weights are the Euclidean distances between those vertices, the Euclidean Leaf-Constrained Minimum Spanning Tree (e-LCMST) problem seeks a minimum cost spanning tree that has at least a specified number, L, of leaf vertices, 2 ≤ L ≤ |V| − 1. The problem is known to be NP-hard, and finds application in facilities location, circuit and network design. Extant heuristics for the problem take O(|V|4) time to compute low-cost leaf-constrained spanning trees, thereby restricting their applicability to only small-sized problem instances. Several meta-heuristic strategies have also been applied for the problem. This paper proposes two novel heuristics that obtain good solutions for the e-LCMST problem in O(|V|3) time and scale well with larger problem sizes. Some relevant theoretical results are also presented. In the course of several computational experiments, the proposed heuristics are shown to outperform the extant heuristics on a large set of benchmark instances for the problem. Results are also reported for the first time on much larger-sized problems than attempted hitherto in the literature. The proposed heuristics are also shown to obtain results that are competitive with those obtained by the best-known extant meta-heuristics for the problem. The computation time taken by the proposed heuristics is well bounded, unlike that for the meta-heuristics, and observed in the experiments to be relatively much lower. Further, the proposed heuristics also have the added advantage of being easily parallelizable, and are shown to obtain good computational speedups in the course of several experiments on large Euclidean benchmark instances.