A Note on a Recent Algorithm for Minimum Cut

TL;DR

This paper simplifies and improves an existing algorithm for finding minimum cuts in graphs, reducing its complexity and replacing a randomized step with a deterministic one, thus enhancing efficiency.

Contribution

It introduces a simpler, faster deterministic algorithm for identifying minimum cuts that 2-respect a spanning tree, improving previous methods.

Findings

Achieved a new complexity of $O(m log^2 n + n log^3 n)$ for the algorithm.

Replaced a randomized step with a deterministic algorithm, simplifying the process.

Enhanced the efficiency of minimum cut algorithms in graph theory.

Abstract

Given an undirected edge-weighted graph $G=(V,E)$ with $m$ edges and $n$ vertices, the minimum cut problem asks to find a subset of vertices $S$ such that the total weight of all edges between $S$ and $V \setminus S$ is minimized. Karger's longstanding $O(m \log^3 n)$ time randomized algorithm for this problem was very recently improved in two independent works to $O(m \log^2 n)$ [ICALP'20] and to $O(m \log^2 n + n\log^5 n)$ [STOC'20]. These two algorithms use different approaches and techniques. In particular, while the former is faster, the latter has the advantage that it can be used to obtain efficient algorithms in the cut-query and in the streaming models of computation. In this paper, we show how to simplify and improve the algorithm of [STOC'20] to $O(m \log^2 n + n\log^3 n)$. We obtain this by replacing a randomized algorithm that, given a spanning tree $T$ of $G$, finds in $O(m \log n+n\log^4 n)$ time a minimum cut of $G$ that 2-respects (cuts two edges of) $T$ with a simple $O(m \log n+n\log^2 n)$ time deterministic algorithm for the same problem.