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We combine the work of Garg and Konemann, and Fleischer with ideas fromdynamic graph algorithms to obtain faster (1-eps)-approximation schemes forvarious versions of the multicommodity flow problem. In particular, if eps ismoderately small and the size of every number used in the input instance ispolynomially bounded, the running times of our algorithms match - up topoly-logarithmic factors and some provably optimal terms - the Omega(mn)flow-decomposition barrier for single-commodity flow.

Faster Approximation Schemes for Fractional Multicommodity Flow Problems via Dynamic Graph Algorithms