A Simple Algorithm for Minimum Cuts in Near-Linear Time

TL;DR

This paper introduces a simple, efficient algorithm for finding minimum cuts in undirected, weighted graphs that simplifies Karger's near-linear time algorithm, making it easier to implement while maintaining optimal complexity.

Contribution

A new straightforward algorithm for minimum cuts that replaces complex subroutines in Karger's method, achieving similar efficiency with simpler implementation.

Findings

Runs in O(m log^3 n) time with high probability

Matches the complexity of Karger's original algorithm

Simplifies the implementation of minimum cut algorithms

Abstract

We consider the minimum cut problem in undirected, weighted graphs. We give a simple algorithm to find a minimum cut that $2$-respects (cuts two edges of) a spanning tree $T$ of a graph $G$. This procedure can be used in place of the complicated subroutine given in Karger's near-linear time minimum cut algorithm (J. ACM, 2000). We give a self-contained version of Karger's algorithm with the new procedure, which is easy to state and relatively simple to implement. It produces a minimum cut on an $m$-edge, $n$-vertex graph in $O(m \log^3 n)$ time with high probability, matching the complexity of Karger's approach.