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.

翻译：在边插入和删除下动态维护图$G$中的最小割是动态图算法中的一个基本问题，目前尚不存在关于每操作时间的条件性下界。在一个$n$节点图中，已知最佳的$(1+o(1))$近似算法需要$\tilde O(\sqrt{n})$的更新时间[Thorup 2007]。若最小割保证为$(\log n)^{o(1)}$，则存在具有$n^{o(1)}$更新时间的确定性精确算法[Jin, Sun, Thorup 2024]。我们提出了首个具有$n^{o(1)}$更新时间的完全动态$(1+o(1))$近似最小割算法。我们的主要技术贡献在于证明只需考虑在适当收缩图中的小容量割即可。