Approximating Fractional Multicommodity Ow Independent of the Number of Commodities

We describe fully polynomial time approximation schemes for various multicommodity ow problems in graphs with m edges and n vertices. We present the rst approximation scheme for maximum multicommodity ow that is independent of the number of commodities k, and our algorithm improves upon the runtime of previous algorithms by this factor of k, performing in O (?2 m 2) time. For maximum concurrent ow, and minimum cost concurrent ow, we present algorithms that are faster than the current known algorithms when the graph is sparse or the number of commodities k is large, i.e. k > m=n. Our algorithms build on the framework proposed by Garg and KK onemann 4]. They are simple, deterministic, and for the versions without costs, they are strongly polynomial. Our maximum multicommodity ow algorithm extends to an approximation scheme for the maximum weighted multicommodity ow, which is faster than those implied by previous algorithms by a factor of k= logW where W is the maximum weight of a commodity.