Recursive Random Contraction Revisited

TL;DR

This paper revisits the recursive random contraction algorithm for minimum cuts in graphs, providing a simplified analysis and establishing bounds on its success probability, while also exploring related variants.

Contribution

It offers a clearer proof of the success probability for the Karger-Stein algorithm on graphs and shows that no similar algorithm can asymptotically improve this probability.

Findings

Success probability bounded below by 1/(2H_n-2) for fixed minimum cuts.

Analysis simplifies when specialized to graphs compared to hypergraphs.

No variant can asymptotically outperform the established success probability.

Abstract

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.