Approximate Minimum-Cost Multicommodity Flows In ... Time

Abstract

We show that an \epsilon-approximate solution of the cost-constrained K-commodity flow problem on an N-node M-arc network G can be computed by sequentially solving O(K(\epsilon^{-2} log K) log M log(\epsilon^{-1}K) single-commodity minimum-cost flow problems on the same network. In particular, an approximate minimumcost multicommodity flow can be computed in O^~(\epsilon^{-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 \epsilon^{-1}. We also provide a simple O^~(NM)-time algorithm for singlecommodity budget-constrained minimum-cost flows which is O^~(\epsilon^{-3}) times faster than the algorithm of Karger and Plotkin (1995)