Minimum cuts in near-linear time

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( m log 3 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.