Fast Approximation Algorithms for Multicommodity Flow Problems

All previously known algorithms for solving the multicommodity ow problem with capacities are based on linear programming. The best of these algorithms [15] uses a fast matrix multiplication algorithm and takes O(k3:5n3m:5 log(nDU )) time for the multicommodity ow problem with integer demands and at least O(k2:5n2m:5 log(n 1DU )) time to nd an approximate solution, where k is the number of commodities, n and m denote the number of nodes and edges in the network, D is the largest demand, and U is the largest edge capacity. Substantially more time is needed to nd an exact solution. As a consequence, even multicommodity ow problems with just a few commodities are believed to be much harder than single-commodity maximumow or minimum-cost ow problems. In this paper, we describe the rst polynomial-time combinatorial algorithms for approximately solving the multicommodity ow problem. The running time of our randomized algorithm is (up to log factors) the same as the time needed to solve k single-commodity ow problems, thus giving the surprising result that approximately computing a k-commodity maximumow is not much harder than computing about k single-commodity maximumows in isolation. In fact, we prove that a (simple) k-commodity ow problem can be approximately solved by approximately solving O(k log2 n) single-commodity minimum-cost ow problems. Our kcommodity algorithm runs in O(knm log4 n) time with high probability. We also describe a deterministic algorithm that uses an O(k)-factor more time. Given any multicommodity ow problem as input, both algorithms are guaranteed to provide a feasible solution to a modi ed ow problem in which all capacities are increased by a (1 + )-factor, or to provide a proof that there is no feasible solution to the original problem. We also describe faster approximation algorithms for multicommodity ow problems with a special structure, such as those that arise in the \sparsest cut" problems studied in [8, 10, 9], and the uniform concurrent ow problems studied in [12, 9] if k pm. 2

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