Vertex connectivity in poly-logarithmic max-flows

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

[1]  Noga Alon,et al.  The space complexity of approximating the frequency moments , 1996, STOC '96.

[2]  Denis Gifford.  1969 , 2018, The British Film Catalogue.

[3]  Aaron Sidford,et al.  Unit Capacity Maxflow in Almost $O(m^{4/3})$ Time , 2020, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS).

[4]  Robert E. Tarjan,et al.  Dividing a Graph into Triconnected Components , 1973, SIAM J. Comput..

[5]  Robert E. Tarjan,et al.  Network Flow and Testing Graph Connectivity , 1975, SIAM J. Comput..

[6]  Fabian Kuhn,et al.  Distributed connectivity decomposition , 2013, PODC.

[7]  Kurt Mehlhorn,et al.  A Probabilistic Algorithm for Vertex Connectivity of Graphs , 1982, Inf. Process. Lett..

[8]  Arkady Kanevsky,et al.  Improved algorithms for graph four-connectivity , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[9]  Yin Tat Lee,et al.  Minimum cost flows, MDPs, and ℓ1-regression in nearly linear time for dense instances , 2021, STOC.

[10]  Andrew V. Goldberg,et al.  A new approach to the maximum flow problem , 1986, STOC '86.

[11]  Loukas Georgiadis,et al.  Testing 2-Vertex Connectivity and Computing Pairs of Vertex-Disjoint s-t Paths in Digraphs , 2010, ICALP.

[12]  László Lovász,et al.  Rubber bands, convex embeddings and graph connectivity , 1988, Comb..

[13]  姜寅求.  1970 , 1970, Literatur in der SBZ/DDR.

[14]  Graham Cormode,et al.  A unifying framework for ℓ0-sampling algorithms , 2013, Distributed and Parallel Databases.

[15]  Thatchaphol Saranurak,et al.  Breaking quadratic time for small vertex connectivity and an approximation scheme , 2019, STOC.

[16]  Liu Yang,et al.  Computing and Testing Small Connectivity in Near-Linear Time and Queries via Fast Local Cut Algorithms , 2019, SODA.

[17]  D. Gifford.  1973: , 1973, Charlotte Delbo.

[18]  E. A. Dinic.  Algorithm for solution of a problem of maximal flow in a network with power estimation , 1970 .

[19]  Di Wang,et al.  Local Flow Partitioning for Faster Edge Connectivity , 2017, SODA.

[20]  Robert E. Tarjan,et al.  Depth-First Search and Linear Graph Algorithms , 1972, SIAM J. Comput..

[21]  James B. Orlin,et al.  A Faster Algorithm for Finding the Minimum Cut in a Directed Graph , 1994, J. Algorithms.

[22]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[23]  Satish Rao,et al.  Computing vertex connectivity: new bounds from old techniques , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[24]  Debmalya Panigrahi,et al.  Deterministic Min-cut in Poly-logarithmic Max-flows , 2020, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS).

[25]  Daniel J. Kleitman,et al.  Methods for investigating the connectivity of large graphs , 1969 .

[26]  Joseph Cheriyan,et al.  Algorithms for parallel k-vertex connectivity and sparse certificates , 1991, STOC '91.