Fast approximation algorithms for multicommodity flow problems

Abstract

All previously known algorithms for solving the multicommodity flow problem with capacities are based on linear programming. The best of these algorithms [14] uses a fast matrix multiplication algorithm and takes O(k25n2m5 log(nDU)) time to find an approximate solution, where κ 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 find an exact solution. As a consequence, even multicommodit y flow problems with jnst a few commodities are believed to be much harder than single-commodity maximum-flow or minimum-cost flow problems. In thk paper, we describe the first polynomial-Time combinatorial algorithms for approximately solving the multicommodity flow problem. The running time of our randomized algorithm is (up to ,log factors) the same as the time needed to solve κ single-commodity flow problems, thus giving the surprising result that approximately computing a κ-commodity maximum-flow is not much harder than computing about κ single-commodity maximum-flows in isolation. In fact, we prove that a (simple) κ-commodity flow problem can be approximately solved by approximately solving O(κ log2 n) single-commodity minimum-cost flow problems. Our κ-commodity algorithm runs in O(knm log4 n) time with high probability. We also describe a deterministic algorithm that uses an O(κ)-factor more time. Given any multicommodit y flow problem as input, both rdgorithms are guaranteed to provide a feasible solution to a modified fIOW problem in which all capacities are increased by a (1 + c)-factor, or to provide a proof that there is no feasible solution to the original problem. We also describe faster approximation algorithms for multicommodity flow problems with a special structure, such as those that arise in the 'sparsest cutn problems studied in [8, 10, 9], and the uniform concurrent flow problems studied in [13, 9] if κ ≥ √m.