Randomized Approximation Schemes for Cuts and Flows in Capacitated Graphs

We improve on random sampling techniques for approximately solving problems that involve cuts and flows in graphs. We give a near-linear-time construction that transforms any graph on vertices into an -edge graph on the same vertices whose cuts have approximately the same value as the original graph’s. In this new graph, for example, we can run the
-time maximum flow algorithm of Goldberg and Rao to find an – minimum cut in
time. This corresponds to a -times minimum – cut in the original graph. In a similar way, we can approximate a sparsest cut to within in
time using a previous
-time algorithm. A related approach leads to a randomized divide and conquer algorithm producing an approximately maximum flow in time.