TY - JOUR

T1 - The geometric generalized minimum spanning tree problem with grid clustering

AU - Feremans, C.

AU - Grigoriev, A.

AU - Sitters, R.

PY - 2006/1/1

Y1 - 2006/1/1

N2 - 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 np\mathcal{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.

AB - 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 np\mathcal{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.

U2 - 10.1007/s10288-006-0012-6

DO - 10.1007/s10288-006-0012-6

M3 - Article

VL - 4

SP - 319

EP - 329

JO - 4 OR - A Quarterly Journal of Operations Research

JF - 4 OR - A Quarterly Journal of Operations Research

SN - 1619-4500

IS - 4

ER -