Faster approximation schemes for fractional multicommodity flow problems

We present fully polynomial approximation schemes for concurrent multicommodity flow problems that run in time of the minimum possible dependencies on the number of commodities k. We show that by modifying the algorithms by Garg and Könemann [1998] and Fleischer [2000], we can reduce their running time on a graph with n vertices and m edges from Õ(ε⁻²(m² + km)) to Õ(ε⁻²m²) for an implicit representation of the output, or Õ(ε⁻²(m² + kn for an explicit representation, where Õ(f) denotes a quantity that is O(f log^O(1)m). The implicit representation consists of a set of trees rooted at sources (there can be more than one tree per source), and with sinks as their leaves, together with flow values for the flow directed from the source to the sinks in a particular tree. Given this implicit representation, the approximate value of the concurrent flow is known, but if we want the explicit flow per commodity per edge, we would have to combine all these trees together, and the cost of doing so may be prohibitive. In case we want to calculate explicitly the solution flow, we modify our schemes so that they run in time polylogarithmic in nk (n is the number of nodes in the network). This is within a polylogarithmic factor of the trivial lower bound of time Ω(nk) needed to explicitly write down a multicommodity flow of k commodities in a network of n nodes. Therefore our schemes are within a polylogarithmic factor of the minimum possible dependencies of the running time on the number of commodities k.