Faster Approximation Algorithms For the Unit Capacity Concurrent Flow Problem with Applications to Routing and Finding Sparse Cuts

Abstract

This paper describes new algorithms for approximately solving the concurrent multicommodity flow problem with uniform capacities. These algorithms are much faster than algorithms discovered previously. Besides being an important problem in its own right, the uniform-capacity concurrent flow problem has many interesting applications. Leighton and Rao used uniform-capacity concurrent flow to find an approximately “sparsest cut” in a graph and thereby approximately solve a wide variety of graph problems, including minimum feedback arc set, minimum cut linear arrangement, and minimum area layout. However, their method appeared to be impractical as it required solving a large linear program. This paper shows that their method might be practical by giving an $O(m^2 \log m)$ expected-time randomized algorithm for their concurrent flow problem on an m-edge graph. Raghavan and Thompson used uniform-capacity concurrent flow to solve approximately a channel width minimization problem in very large scale integration. An $O(k^{{3 / 2}} (m + n\log n)$ expected-time randomized algorithm and an $O(k\min \{ n,k\} (m + n\log n)\log k)$ deterministic algorithm is given for this problem when the channel width is $\Omega (\log n)$, where k denotes the number of wires to be routed in an n-node, m-edge network.