Min Cut , Fast Cut , Polynomial Identities

Throughout this section, G = (V,E) is a multi-graph. A cut of G is a bipartition (S, S̄) of the vertex set of G. The capacity of a cut is the number of edges having one endpoint on both sides of the cut. A min-cut is a cut of minimum capacity. A minimum cut can be computed with the help of maxflow computations. For some vertex s and every other vertex t, one computes the minimum cut separating s from t, and then takes the smallest cut obtained in this way. There are better deterministic algorithms, see for example [SW97] and [MN99, Section 7.12] Karger and Stein [KS96] found a simple and efficient randomized algorithm. We will introduce their algorithm in two steps. We first describe an extremely simple algorithms that finds the minimum cut with a nonzero, but small probability. In a second step, we will show how to increase the success probability. The resulting algorithm is a Monte Carlo algorithm. It finds the minimum cut with good probability, but is not guaranteed to find the minimum cut.