1 Introduction Consider a complete graph G(V; E) with node set V = fv1; v2; : : : ; vn g. In the problem which we consider, both edges and nodes are assigned numbers to reflect their lengths and weights. Associated with each edge (vi; vj) is a non-negative number d(i; j) that represents its length. Similarly, associated with each node vi is a non-negative number wi that represents its weight. Let W be the sum of the weights of all nodes