Randomized Approximation Schemes for Cuts and Flows in Capacitated Graphs
<|reference_start|>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 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.<|reference_end|>
@article{benczur2002randomized,
    title={Randomized Approximation Schemes for Cuts and Flows in Capacitated
  Graphs},
    author={Andras Benczur and David R. Karger},
    journal={arXiv preprint arXiv:cs/0207078},
    year={2002},
    archivePrefix={arXiv},
    eprint={cs/0207078},
    primaryClass={cs.DS cs.DM}
}