Minimum Cuts in Near-Linear Time
<|reference_start|>Minimum Cuts in Near-Linear Time: We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a ``semi-duality'' between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized algorithm that finds a minimum cut in an m-edge, n-vertex graph with high probability in O(m log^3 n) time. We also give a simpler randomized algorithm that finds all minimum cuts with high probability in O(n^2 log n) time. This variant has an optimal RNC parallelization. Both variants improve on the previous best time bound of O(n^2 log^3 n). Other applications of the tree-packing approach are new, nearly tight bounds on the number of near minimum cuts a graph may have and a new data structure for representing them in a space-efficient manner.<|reference_end|>
@article{karger1998minimum,
    title={Minimum Cuts in Near-Linear Time},
    author={David R. Karger},
    journal={arXiv preprint arXiv:cs/9812007},
    year={1998},
    archivePrefix={arXiv},
    eprint={cs/9812007},
    primaryClass={cs.DS}
}