We present fully polynomial approximation schemes for concurrent multicommodity flow prob-lems that run in time of minimum possible dependency on the number of commodities k. We showthat by modifying the algorithms by Garg & K&quot;onemann  and Fleischer  we can reduce their running time on a graph with n vertices and m edges from ~O(&quot;-2(m2 + km)) to ~O(&quot;-2m2) foran implicit representation of the output, or ~ O(&quot;-2(m2 + kn)) for an explicit representation, where ~ O(f) denotes a quantity that is O(f logO(1) m). The implicit representation consists of a set oftrees 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 poly-logarithmic in nk (n is the numberof nodes in the network). This is within a poly-logarithmic factor of the trivial lower bound of time \Omega (nk) needed to explicitly write down a multicommodity flow of k commodities in a network of n nodes. Therefore our schemes are within a poly-logarithmic factor of the minimum possible dependency of the running time on the number of commodities k
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