Minimum Cut in $O(m\log^2 n)$ Time

TL;DR

This paper presents a faster randomized algorithm for finding minimum cuts in graphs, improving the previous best time complexity from $O(m \, \log^3 n)$ to $O(m \, \log^2 n)$, with a key technical contribution involving a deterministic subroutine.

Contribution

It introduces the first algorithm to improve the minimum cut computation time to $O(m \, \log^2 n)$, advancing the state-of-the-art since 1996.

Findings

Achieved $O(m \log^2 n)$ time complexity for minimum cut detection.

Developed a deterministic $O(m \log n)$ algorithm for 2-respecting cuts.

First improvement over Karger's $O(m \log^3 n)$ algorithm since 1996.

Abstract

We give a randomized algorithm that finds a minimum cut in an undirected weighted $m$-edge $n$-vertex graph $G$ with high probability in $O(m \log^2 n)$ time. This is the first improvement to Karger's celebrated $O(m \log^3 n)$ time algorithm from 1996. Our main technical contribution is a deterministic $O(m \log n)$ time algorithm that, given a spanning tree $T$ of $G$, finds a minimum cut of $G$ that 2-respects (cuts two edges of) $T$.