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Vertex connectivity in poly-logarithmic max-flows

Authors:

Danupon Nanongkai,

Debmalya Panigrahi,

Thatchaphol Saranurak,

Sorrachai YingchareonthawornchaiAuthors Info & Claims

STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

Pages 317 - 329

https://doi.org/10.1145/3406325.3451088

Published: 15 June 2021 Publication History

Abstract

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 (m^α) time for any α ≥ 1, if there is a m^α-time maxflow algorithm. Using the current best maxflow algorithm that runs in m^4/3+o(1) time (Kathuria, Liu and Sidford, FOCS 2020), this yields a m^4/3+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 mn^{1−1/12+o(1)} time

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Cited By

Chen LKyng RLiu YPeng RGutenberg MSachdeva S(2023)Almost-Linear-Time Algorithms for Maximum Flow and Minimum-Cost FlowCommunications of the ACM10.1145/361094066:12(85-92)Online publication date: 17-Nov-2023
https://dl.acm.org/doi/10.1145/3610940
Brand JChen LKyng RLiu YPeng RGutenberg MSachdeva SSidford A(2023)A Deterministic Almost-Linear Time Algorithm for Minimum-Cost Flow2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00037(503-514)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00037
Parter MPetruschka A(2023)Near-optimal distributed computation of small vertex cutsDistributed Computing10.1007/s00446-023-00455-z37:2(67-88)Online publication date: 14-Jul-2023
https://dl.acm.org/doi/10.1007/s00446-023-00455-z
Show More Cited By

Index Terms

Vertex connectivity in poly-logarithmic max-flows
1. Theory of computation
  1. Design and analysis of algorithms
    1. Graph algorithms analysis

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cover image ACM Conferences

STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

June 2021

1797 pages

ISBN:9781450380539

DOI:10.1145/3406325

General Chair:
Samir Khuller
Northwestern University, USA
,
Program Chair:
Virginia Vassilevska Williams
Massachusetts Institute of Technology, USA

Copyright © 2021 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

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Published: 15 June 2021

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STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing

June 21 - 25, 2021

Virtual, Italy

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Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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Cited By

Chen LKyng RLiu YPeng RGutenberg MSachdeva S(2023)Almost-Linear-Time Algorithms for Maximum Flow and Minimum-Cost FlowCommunications of the ACM10.1145/361094066:12(85-92)Online publication date: 17-Nov-2023
https://dl.acm.org/doi/10.1145/3610940
Brand JChen LKyng RLiu YPeng RGutenberg MSachdeva SSidford A(2023)A Deterministic Almost-Linear Time Algorithm for Minimum-Cost Flow2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00037(503-514)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00037
Parter MPetruschka A(2023)Near-optimal distributed computation of small vertex cutsDistributed Computing10.1007/s00446-023-00455-z37:2(67-88)Online publication date: 14-Jul-2023
https://dl.acm.org/doi/10.1007/s00446-023-00455-z
Franck MYingchareonthawornchai S(2022)Engineering Nearly Linear-time Algorithms for Small Vertex ConnectivityACM Journal of Experimental Algorithmics10.1145/356482227(1-29)Online publication date: 13-Dec-2022
https://dl.acm.org/doi/10.1145/3564822
Blikstad JVan Den Brand JEfron YMukhopadhyay SNanongkai D(2022)Nearly Optimal Communication and Query Complexity of Bipartite Matching2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00113(1174-1185)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00113
Abboud AKrauthgamer RLi JPanigrahi DSaranurak TTrabelsi O(2022)Breaking the Cubic Barrier for All-Pairs Max-Flow: Gomory-Hu Tree in Nearly Quadratic Time2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00088(884-895)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00088
Saranurak TYingchareonthawornchai S(2022)Deterministic Small Vertex Connectivity in Almost Linear Time2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00080(789-800)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00080
Cen RLi JNanongkai DPanigrahi DSaranurak TQuanrud K(2022)Minimum Cuts in Directed Graphs via Partial Sparsification2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00113(1147-1158)Online publication date: Feb-2022
https://doi.org/10.1109/FOCS52979.2021.00113

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