A Simple Algorithm for Minimum Cuts in Near-Linear Time

We consider the minimum cut problem in undirected, weighted graphs. We give a simple algorithm to find a minimum cut that \(2\)-respects (cuts two edges of) a spanning tree \(T\) of a graph \(G\). This procedure can be used in place of the complicated subroutine given in Karger's near-linear time minimum cut algorithm (J. ACM, 2000). We give a self-contained version of Karger's algorithm with the new procedure, which is easy to state and relatively simple to implement. It produces a minimum cut on an \(m\)-edge, \(n\)-vertex graph in \(O(m \log^3 n)\) time with high probability, matching the complexity of Karger's approach.