Fully Dynamic Approximate Minimum Cut in Subpolynomial Time per Operation

TL;DR

This paper introduces the first fully dynamic algorithm for approximate minimum cut in graphs with subpolynomial update time, advancing the efficiency of dynamic graph algorithms.

Contribution

It presents a novel approach that reduces the problem to considering small-volume cuts in contracted graphs, enabling faster updates.

Findings

Achieves subpolynomial update time for approximate minimum cut

Introduces a new technique focusing on small-volume cuts

Advances the state-of-the-art in dynamic graph algorithms

Abstract

Dynamically maintaining the minimum cut in a graph $G$ under edge insertions and deletions is a fundamental problem in dynamic graph algorithms for which no conditional lower bound on the time per operation exists. In an $n$-node graph the best known $(1+o(1))$-approximate algorithm takes $\tilde O(\sqrt{n})$ update time [Thorup 2007]. If the minimum cut is guaranteed to be $(\log n)^{o(1)}$, a deterministic exact algorithm with $n^{o(1)}$ update time exists [Jin, Sun, Thorup 2024]. We present the first fully dynamic algorithm for $(1+o(1))$-approximate minimum cut with $n^{o(1)}$ update time. Our main technical contribution is to show that it suffices to consider small-volume cuts in suitably contracted graphs.