Finding a Small Vertex Cut on Distributed Networks

We present an algorithm for distributed networks to efficiently find a small vertex cut in the CONGEST model. Given a positive integer $\kappa$, our algorithm can, with high probability, either find $\kappa$ vertices whose removal disconnects the network or return that such $\kappa$ vertices do not exist. Our algorithm takes $\kappa^3\cdot \tilde{O}(D+\sqrt{n})$ rounds, where $n$ is the number of vertices in the network and $D$ denotes the network’s diameter. This implies $\tilde{O}(D+\sqrt{n})$ round complexity whenever $\kappa= \text{poly} \log(n)$.

Prior to our result, a similar bound of $\tilde{O}(D+\sqrt{n})$ is known only when $\kappa=1$ [Thurimella PODC'95]. For $\kappa\geq 2$, this bound can be obtained only by an $O(\log n)$-approximation algorithm [Censor-Hillel, Ghaffari, Kuhn PODC'14], and the only known exact algorithm takes $O\left((\kappa\Delta D)^{O(\kappa)}\right)$ rounds, where $\Delta$ is the maximum degree [Parter DISC'19]. Our result answers an open problem by Nanongkai, Saranurak, and Yingchareonthawornchai [STOC'19].