IMPROVED BOUNDS FOR THE MAX-FLOW MIN-MULTICUT RATIO FOR PLANAR AND K(R,R)-FREE GRAPHS

We consider the version of the multicommodity flow problem in which the objective is to maximize the sum of commodities routed. Garg, Vazirani and Yannakakis proved that the minimum multicut and maximum flow ratio for this problem can be bounded by O(log k), where k is the number of commodities. In this note we improve this ratio to O(1) for planar graphs, and more generally to O(r3) for graphs with an excluded K(r,r) minor. The proof is based on the network decomposition theorem of Klein, Plotkin and Rao. Our proof is constructive and yields approximation algorithms, with the same factors, for the minimum multicut problem on such networks.