An Approximation Scheme for the Generalized Geometric Minimum Spanning Tree Problem with Grid Clustering

This paper is concerned with a special case of the Generalized Minimum Spanning Tree Problem. The Generalized Minimum Spanning Tree Problem is defined on an undirected graph, where the vertex set is partitioned into clusters, and non-negative costs are associated with the edges. The problem is to find a tree of minimum cost containing exactly one vertex in each cluster. We consider a geometric case of the problem where the graph is complete, all vertices are situated in the plane, and Euclidean distance defines the edge cost. We prove that the problem admits PTAS if restricted to grid clustering.