IMPROVED BOUNDS ON THE MAX-FLOW MIN-CUT RATIO FOR MULTICOMMODITY FLOWS

In this paper we consider the worst case ratio between the capacity of min-cuts and the value of max-flow for multicommodity how problems. We improve the best known bounds for the min-cut max-flow ratio for multicommodity flows in undirected graphs, by replacing the O(log D) in the bound by O(log k), where D denotes the sum of all demands, and k denotes the number of commodities. In essence we prove that up to constant factors the worst min-cut max-how ratios appear in problems where demands are integral and polynomial in the number of commodities. Klein, Rao, Agrawal, and Ravi have previously proved that if the demands and the capacities are integral, then the min-cut max-flow ratio in general undirected graphs is bounded by O(log C log D), where C denotes the sum of all the capacities. Tragoudas has improved this bound to O(log n log D), where n is the number of nodes in the network. Garg, Vazirani and Yannakakis further improved this to O(log k log D). Klein, Plotkin and Rao have proved that for planar networks, the ratio is O(log D). Our result improves the bound for general networks to O(log(2) k) and the bound for planar networks to O(log k). In both cases our result implies the first non-trivial bound that is independent of the magnitude of the numbers involved. The method presented in this paper can be used to give polynomial time approximation algorithms to the minimum cuts in the network up to the above factors.