Tree-Approximations for the Weighted Cost-Distance Problem

We generalize the Cost-Distance problem: Given a set of sites in -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 any given set of sites and a non-negative 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 . Finally, we show that better approximation rates are not possible for trees. We prove this by giving a counter-example. Thus, we show that for this instance that every tree solution differs from the optimal network by a factor .