Approximate Minimum - Cost Multicommodity Flows in ~ O ( " � 2 Knm ) Time

We show that an "-approximate solution of the cost-constrained K-commodity ow problem on an N-nodeM-arc network G can be computed by sequentially solving O K(" 2 + logK) logM log(" 1 K) single-commodity minimum-cost ow problems on the same network. In particular, an approximate minimumcost multicommodity ow can be computed in ~ O(" 2 KNM) running time, where the notation ~ O( ) means \up to logarithmic factors". This result improves the time bound mentioned in Grigoriadis and Khachiyan (1994) by a factor of M=N and that developed recently in Karger and Plotkin (1995) by a factor of " 1 . We also provide a simple ~ O (NM)-time algorithm for singlecommodity budget-constrained minimum-cost ows which is ~ O " 3 times faster than the algorithm of Karger and Plotkin (1995). APPROXIMATE MINIMUM-COST MULTICOMMODITY FLOWS IN ~ O(" 2 KNM) TIME* Michael D. Grigoriadis and Leonid G. Khachiyan Department of Computer Science, Rutgers University, New Brunswick, NJ, USA