I just want to show the idea of this algorithm. I show this algorithm for the multi-commodity flow problem, so we have a set of k Pairs and we want to route as much flow as possible between any pair.
What we do is to formulate the dual problem that is a cut problem. We want to minimize the sum over all the edges of the length of the edges times the cost of the edges under the assumption that any pair of edges is far away for at most a value 1. Moreover, we define the minimum for any length function, so, for any solution not yet feasible for the dual problem a value a(l). a(l) is the minimum distance between two vertices of a pair. And we can see that our problem reduces to finding the minimum length function l such that the ratio between the total volume and a l is minimized.