On Shortest Networks for Classes of Points in the Plane

We are given a set P of points in the plane, together with a partition of P into classes of points; i.e., each point of P belongs to exactly one class. For a given network optimization problem, such as nding a minimum spanning tree or nding a minimum diameter spanning tree, we study the problem of choosing a subset P 0 of P that contains at least one point of each class and solving the network optimization problem for P 0 , such that the solution is optimal among all possible choices for P 0. We show that solving the minimum spanning tree problem for classes of points in the plane is NP-complete, where the distance between points is deened by any of the Minkowski metrics L p ; 1 p 1. In contrast, a class solution for the minimum diameter spanning tree problem can be computed in time O(jPj 3). By proving the NP-completeness of the minimum spanning tree class problem we also get some results for distance graphs. Here, computing a class solution for the minimum spanning tree problem is NP-complete, even under several restrictions, e.g., if the graph is part of a unit grid and is a tree, where the vertex degree and the number of vertices per class are both bounded by three. This is true even if the graph is a minimum spanning tree for its set of vertices.