# On shortest networks for classes of points in the plane

## Abstract

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 finding a minimum spanning tree or finding a minimum diameter spanning tree, we study the problem of choosing a subset *P′* of *P* that contains at least one point of each class and solving the network optimization problem for *P′*, such that the solution is optimal among all possible choices for *P′*. 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 defined by any of the Minkowski metrics *L*_{ p }, 1⩽*p*⩽∞. In contrast, a class solution for the minimum diameter spanning tree problem can be computed in time *O*(¦P¦^{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.

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