research-article

Deterministic Global Minimum Cut of a Simple Graph in Near-Linear Time

Authors:

Ken-ichi Kawarabayashi,

Mikkel ThorupAuthors Info & Claims

STOC '15: Proceedings of the forty-seventh annual ACM symposium on Theory of Computing

Pages 665 - 674

https://doi.org/10.1145/2746539.2746588

Published: 14 June 2015 Publication History

Abstract

We present a deterministic near-linear time algorithm that computes the edge-connectivity and finds a minimum cut for a simple undirected unweighted graph G with n vertices and m edges. This is the first o(mn) time deterministic algorithm for the problem. In near-linear time we can also construct the classic cactus representation of all minimum cuts.

The previous fastest deterministic algorithm by Gabow from STOC'91 took O(m+λ² n), where λ is the edge connectivity, but λ could be Ω(n).

At STOC'96 Karger presented a randomized near linear time Monte Carlo algorithm for the minimum cut problem. As he points out, there is no better way of certifying the minimality of the returned cut than to use Gabow's slower deterministic algorithm and compare sizes.

Our main technical contribution is a near-linear time algorithm that contracts vertex sets of a simple input graph G with minimum degree δ, producing a multigraph G with ~O(m/δ) edges which preserves all minimum cuts of G with at least two vertices on each side.

In our deterministic near-linear time algorithm, we will decompose the problem via low-conductance cuts found using PageRank a la Brin and Page (1998), as analyzed by Andersson, Chung, and Lang at FOCS'06. Normally such algorithms for low-conductance cuts are randomized Monte Carlo algorithms, because they rely on guessing a good start vertex. However, in our case, we have so much structure that no guessing is needed.

References

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Cited By

Fox KPanigrahi DZhang F(2023)Minimum Cut and Minimum k-Cut in Hypergraphs via Branching ContractionsACM Transactions on Algorithms10.1145/357016219:2(1-22)Online publication date: 15-Apr-2023
https://dl.acm.org/doi/10.1145/3570162
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
Chen LPeng RWang D(2022)2-norm Flow Diffusion in Near-Linear Time2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00060(540-549)Online publication date: Feb-2022
https://doi.org/10.1109/FOCS52979.2021.00060
Show More Cited By

Index Terms

Deterministic Global Minimum Cut of a Simple Graph in Near-Linear Time
1. Theory of computation
  1. Randomness, geometry and discrete structures

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cover image ACM Conferences

STOC '15: Proceedings of the forty-seventh annual ACM symposium on Theory of Computing

June 2015

916 pages

ISBN:9781450335362

DOI:10.1145/2746539

General Chair:
Rocco Servedio
Columbia University
,
Program Chair:
Ronitt Rubinfeld
MIT and Tel Aviv University

Copyright © 2015 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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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 14 June 2015

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Funding Sources

JST
Mitsubishi Foundation
the Danish Council for Independent Research under the Sapere Aude research career programme

Conference

STOC '15

Sponsor:

SIGACT

STOC '15: Symposium on Theory of Computing

June 14 - 17, 2015

Oregon, Portland, USA

Acceptance Rates

STOC '15 Paper Acceptance Rate 93 of 347 submissions, 27%;

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

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STOC '25

Sponsor:
sigact

57th Annual ACM Symposium on Theory of Computing (STOC 2025)

June 23 - 27, 2025

Prague , Czech Republic

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Citations

Cited By

Fox KPanigrahi DZhang F(2023)Minimum Cut and Minimum k-Cut in Hypergraphs via Branching ContractionsACM Transactions on Algorithms10.1145/357016219:2(1-22)Online publication date: 15-Apr-2023
https://dl.acm.org/doi/10.1145/3570162
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
Chen LPeng RWang D(2022)2-norm Flow Diffusion in Near-Linear Time2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00060(540-549)Online publication date: Feb-2022
https://doi.org/10.1109/FOCS52979.2021.00060
Beideman CChandrasekaran KMukhopadhyay SNanongkai D(2022)Faster Connectivity in Low-Rank Hypergraphs via Expander DecompositionInteger Programming and Combinatorial Optimization10.1007/978-3-031-06901-7_6(70-83)Online publication date: 27-May-2022
https://doi.org/10.1007/978-3-031-06901-7_6
Geissmann BGianinazzi L(2021)Parallel Minimum Cuts in Near-linear Work and Low DepthACM Transactions on Parallel Computing10.1145/34608908:2(1-20)Online publication date: 23-Aug-2021
https://dl.acm.org/doi/10.1145/3460890
Fox KPanigrahi DZhang FChan T(2019)Minimum cut and minimum k-Cut in hypergraphs via branching contractionsProceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3310435.3310489(881-896)Online publication date: 6-Jan-2019
https://dl.acm.org/doi/10.5555/3310435.3310489
Chang YPettie SZhang HChan T(2019)Distributed triangle detection via expander decompositionProceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3310435.3310486(821-840)Online publication date: 6-Jan-2019
https://dl.acm.org/doi/10.5555/3310435.3310486
Henzinger MNoe ASchulz C(2019)Shared-Memory Exact Minimum Cuts2019 IEEE International Parallel and Distributed Processing Symposium (IPDPS)10.1109/IPDPS.2019.00013(13-22)Online publication date: May-2019
https://doi.org/10.1109/IPDPS.2019.00013
Williamson D(2019)Network Flow Algorithms10.1017/9781316888568Online publication date: 21-Aug-2019
https://doi.org/10.1017/9781316888568
Gupta RKarmakar S(2019)Incremental Algorithm for Minimum Cut and Edge Connectivity in HypergraphCombinatorial Algorithms10.1007/978-3-030-25005-8_20(237-250)Online publication date: 10-Jul-2019
https://doi.org/10.1007/978-3-030-25005-8_20
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