4OR (2006) 4:319–329 DOI 10.1007/s10288-006-0012-6 REGULAR PAPER The geometric generalized minimum spanning tree problem with grid clustering Corinne Feremans · Alexander Grigoriev · René Sitters Received: 1 June 2005 / Revised: 1 February 2006 / Published online: 19 July 2006 © Springer-Verlag 2006 Abstract This paper is concerned with a special case of the generalized minimum spanning tree problem. The 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 at least 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 is strongly NP -hard even in the case of a special structure of the clustering called grid clustering. We construct an exact exponential time dynamic programming algorithm and, based on this dynamic programming algorithm, we develop a polynomial time approximation scheme for the problem with grid clustering. Keywords Generalized minimum spanning tree · Complexity · Approximations · Grid clustering MSC classification 68Q17 · 68W25 C. Feremans · A. Grigoriev (B ) Department of Quantitative Economics, Maastricht University, P.O. Box 616, 6200 MD, Maastricht, The Netherlands e-mail: a.grigoriev@ke.unimaas.nl C. Feremans e-mail: c.feremans@ke.unimaas.nl R. Sitters Max-Planck-Institute for Computer Science, Saarbrücken, Germany e-mail: sitters@mpi-inf.mpg.de