A new probabilistic analysis of Karger's randomized algorithm for minimum cut problems

Recently Karger proposed a new randomized algorithm for finding a minimum cut of an n-vertex graph (weighted or unweighted) with probability Ω(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, (log n) n ⩽ p ⩽ 1 , we show that the expectation of success probability of the algorithm is Ω( p n ) . We also investigate a class of graphs with special structure that consists of two n- cliques and γ( n − 1) edges between the two cliques. Here γ is a parameter satisfying 0 < γ < 1 that makes these γ( n − 1) edges a unique minimum cut. We show that the algorithm finds the unique minimum cut with probability Ω( γ n γ ) .