We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a "semiduality" between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized (Monte Carlo) 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 ᏺᏯ 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.
This improved on the O(m log 3 n) algorithm of Karger REF and is the state of the art time complexity for edge connectivity.
Minimum cuts in near-linear time
JACM
Computer Science Mathematics