Fully Dynamic Approximate Minimum Cut in Subpolynomial Time per Operation
Antoine El-Hayek, Monika Henzinger, Jason Li
TL;DR
This work resolves a major open question by giving the first fully dynamic, Monte-Carlo algorithm that maintains a $(1+o(1))$-approximate minimum cut in unweighted graphs with $n^{o(1)}$ amortized update time, valid for all min-cut values. The approach combines a Karger-style sparsification with a dynamic expander-based cluster decomposition, a hierarchy of contracted graphs, and mirror clusters to ensure that all relevant minimum cuts are captured as local cuts at some level. Central to the technique are: (i) dynamic expander decomposition, (ii) $(1-\epsilon)$-boundary sparseness, (iii) LocalKCut for finding small cuts within clusters, and (iv) a mirror-cluster mechanism to uncross cuts efficiently. Together, these components yield a provably correct, near-polynomially efficient algorithm with provable high-probability guarantees, marking a substantial advance in dynamic graph algorithms and enabling near-linear per-operation scalability for min-cut maintenance. The results leverage subpolynomial time bounds, an $h$-level cluster hierarchy with depth $O(\log^{1/4} n)$, and multi-scale sparsification to handle all possible min-cut values with controlled recourse and running time.
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.