Recursive Random Contraction Revisited

In this note, we revisit the recursive random contraction algorithm of Karger and Stein for finding a minimum cut in a graph. Our revisit is occasioned by a paper of Fox, Panigrahi, and Zhang which gives an extension of the Karger-Stein algorithm to minimum cuts and minimum $k$-cuts in hypergraphs. When specialized to the case of graphs, the algorithm is somewhat different than the original Karger-Stein algorithm. We show that the analysis becomes particularly clean in this case: we can prove that the probability that a fixed minimum cut in an $n$ node graph is returned by the algorithm is bounded below by $1/(2H_n-2)$, where $H_n$ is the $n$th harmonic number. We also consider other similar variants of the algorithm, and show that no such algorithm can achieve an asymptotically better probability of finding a fixed minimum cut.