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 $κ$, our algorithm can, with high probability, either find $κ$ vertices whose removal disconnects the network or return that such $κ$ vertices do not exist. Our algorithm takes $κ^3\cdot \tilde{O}(D+\sqrt{n})$ 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 $κ=\text{polylog}(n)$. Prior to our result, a bound of $\tilde{O}(D)$ is known only when $κ=1,2$ [Parter, Petruschka DISC'22]. For $κ\geq 3$, 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 $O\left((κΔD)^{O(κ)}\right)$ rounds, where $Δ$ is the maximum degree [Parter DISC'19]. Our result answers an open problem by Nanongkai, Saranurak, and Yingchareonthawornchai [STOC'19].