A New Probabilistic Analysis of Karger's Randomized Algorithm for Minimum Cut Problems

Recently Karger proposed a new randomized algorithm for nding a minimum cut of an n-vertex graph (weighted or unweighted) with probability (n 2 ). In this paper we present a new probabilistic analysis of Karger's randomized algorithm for a few classes of unweighted graphs. For random graphs whose edges are selected with a given probability p, logn n p 1, we show that the expectation of success probability of the algorithm is ( p n ). We also investigate a class of graphs with special structure that consists of two n-cliques and (n 1) edges between the two cliques. Here is a parameter satisfying 0 < < 1 that makes these (n 1) edges a unique minimum cut. We show that the algorithm nds the unique minimum cut with probability ( n ).