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 n vertices into an O(n\log n)-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 O(m^{3/2})-time maximum flow algorithm of Goldberg and Rao to find an s--t minimum cut in O(n^{3/2}) time. This corresponds to a (1+epsilon)-times minimum s--t cut in the original graph. In a similar way, we can approximate a sparsest cut to within O(log n) in O(n^2) time using a previous O(mn)-time algorithm. A related approach leads to a randomized divide and conquer algorithm producing an approximately maximum flow in O(m sqrt{n}) time.