On the Max-Flow Min-Cut Ratio for Directed Multicommodity Flows

We give a pure combinatorial problem whose solution determines max-flow min-cut ratio for directed multicommodity flows. In addition, this combinatorial problem has applications in improving the approximation factor of Greedy algorithm for maximum edge disjoint path problem. More precisely, our upper bound improves the approximation factor for this problem to O(n^{3/4}). Finally, we demonstrate how even for very simple graphs the aforementioned ratio might be very large