# Tree-Approximations for the Weighted Cost-Distance Problem

## Abstract

We generalize the Cost-Distance problem: Given a set of *n* sites in *k*-dimensional Euclidean space and a weighting over pairs of sites, construct a network that minimizes the cost (i.e. weight) of the network and the weighted distances between all pairs of sites. It turns out that the optimal solution can contain Steiner points as well as cycles. Furthermore, there are instances where crossings optimize the network. We then investigate how trees can approximate the weighted Cost-Distance problem. We show that for anyg iven set of *n* sites and a nonnegative weighting of pairs, provided the sum of the weights is polynomial, one can construct in polynomial time a tree that approximates the optimal network within a factor of *O*(log *n*). Finally, we show that better approximation rates are not possible for trees. We prove this bygiv ing a counter-example. Thus, we show that for this instance that everytre e solution differs from the optimal network bya factor ω(log *n*).

## Keywords

Polynomial Time Span Tree Minimum Span Tree Steiner Tree Optimal Network## Preview

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