Vertex Connectivity in Poly-Logarithmic Max-Flows

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Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021
The vertex connectivity of an m-edge n-vertex undirected graph is the smallest number of vertices whose removal disconnects the graph, or leaves only a singleton vertex. In this paper, we give a reduction from the vertex connectivity problem to a set of maxflow instances. Using this reduction, we can solve vertex connectivity in (mα) time for any α ≥ 1, if there is a mα-time maxflow algorithm. Using the current best maxflow algorithm that runs in m4/3+o(1) time (Kathuria, Liu and Sidford, FOCS 2020), this yields a m4/3+o(1)-time vertex connectivity algorithm. This is the first improvement in the running time of the vertex connectivity problem in over 20 years, the previous best being an Õ(mn)-time algorithm due to Henzinger, Rao, and Gabow (FOCS 1996). Indeed, no algorithm with an o(mn) running time was known before our work, even if we assume an (m)-time maxflow algorithm. Our new technique is robust enough to also improve the best Õ(mn)-time bound for directed vertex connectivity to mn1−1/12+o(1) time
| openaire: EC/H2020/759557/EU//ALGOCom
vertex connectivity, algorithmic graph theory
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Li , J , Nanongkai , D , Panigrahi , D , Saranurak , T & Yingchareonthawornchai , S 2021 , Vertex Connectivity in Poly-Logarithmic Max-Flows . in S Khuller & V V Williams (eds) , Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing . STOC 2021 , ACM , New York, NY, USA , pp. 317–329 , ACM Symposium on Theory of Computing , Virtual, Online , 21/06/2021 .