Clustering-based heuristic approaches for the Euclidean Leaf-constrained Minimum Spanning Tree Problem

Given a graph G (V, E) having vertices as points in the two-dimensional Euclidean space and edge-weights representing the Euclidean distances between those vertices, the Euclidean Leaf-Constrained Minimum Spanning Tree (LCMST) problem searches for a minimum cost spanning tree that has at least a specified number of leaf vertices L, 2 ≤ L ≤ |V| − 1. The problem is NP-hard and finds applications in facilities location, circuit, network design, and other domains also. The Euclidean LCMST (e-LCMST) problem arises when the domain is restricted to the Euclidean space. A well-known greedy spanning tree construction-based heuristic for the problem takes O(|V|⁴) time to compute low-cost LCMST but is restricted in its applicability to only small-sized instances. A recent less greedy heuristic for the e-LCMST problem obtains superior performance in O(|V|³) time. This paper proposes four clustering-based heuristics that obtain good solutions for the e-LCMST problem in O(|V|³) time. In the course of several experiments on standard and large complete graph instances, the proposed heuristics are shown to outperform the existing heuristics for the problem.