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Flows in almost linear time via adaptive preconditioning

Authors:

Sushant Sachdeva,

Di WangAuthors Info & Claims

STOC 2019: Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

Pages 902 - 913

https://doi.org/10.1145/3313276.3316410

Published: 23 June 2019 Publication History

Abstract

We present algorithms for solving a large class of flow and regression problems on unit weighted graphs to (1 + 1 / poly(n)) accuracy in almost-linear time. These problems include ℓ_p-norm minimizing flow for p large (p ∈ [ω(1), o(log^2/3 n) ]), and their duals, ℓ_p-norm semi-supervised learning for p close to 1.

As p tends to infinity, p-norm flow and its dual tend to max-flow and min-cut respectively. Using this connection and our algorithms, we give an alternate approach for approximating undirected max-flow, and the first almost-linear time approximations of discretizations of total variation minimization objectives.

Our framework is inspired by the routing-based solver for Laplacian linear systems by Spielman and Teng (STOC ’04, SIMAX ’14), and is based on several new tools we develop, including adaptive non-linear preconditioning, tree-routings, and (ultra-)sparsification for mixed ℓ₂ and ℓ_p norm objectives.

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Cited By

Chen LKyng RLiu YMeierhans SProbst Gutenberg MMohar BShinkar IO'Donnell R(2024)Almost-Linear Time Algorithms for Incremental Graphs: Cycle Detection, SCCs, s-t Shortest Path, and Minimum-Cost FlowProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649745(1165-1173)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649745
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
Show More Cited By

Index Terms

Flows in almost linear time via adaptive preconditioning
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Network flows
  2. Mathematical analysis
    1. Mathematical optimization
      1. Continuous optimization
        Convex optimization
2. Theory of computation
  1. Design and analysis of algorithms
    1. Graph algorithms analysis
      1. Network flows
      2. Sparsification and spanners
    2. Mathematical optimization
      1. Continuous optimization
        Convex optimization

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cover image ACM Conferences

STOC 2019: Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

June 2019

1258 pages

ISBN:9781450367059

DOI:10.1145/3313276

General Chair:
Moses Charikar
Stanford University
,
Program Chair:
Edith Cohen
Google, USA / Tel Aviv University, Israel

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Published: 23 June 2019

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STOC '19: 51st Annual ACM SIGACT Symposium on the Theory of Computing

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Cited By

Chen LKyng RLiu YMeierhans SProbst Gutenberg MMohar BShinkar IO'Donnell R(2024)Almost-Linear Time Algorithms for Incremental Graphs: Cycle Detection, SCCs, s-t Shortest Path, and Minimum-Cost FlowProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649745(1165-1173)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649745
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
van den Brand JGao YJambulapati ALee YLiu YPeng RSidford ALeonardi SGupta A(2022)Faster maxflow via improved dynamic spectral vertex sparsifiersProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3520068(543-556)Online publication date: 9-Jun-2022
https://dl.acm.org/doi/10.1145/3519935.3520068
Jambulapati ALiu YSidford ALeonardi SGupta A(2022)Improved iteration complexities for overconstrained p-norm regressionProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3519971(529-542)Online publication date: 9-Jun-2022
https://dl.acm.org/doi/10.1145/3519935.3519971
Kathuria TLiu YSidford A(2022)Unit Capacity Maxflow in Almost $m^{4/3}$ TimeSIAM Journal on Computing10.1137/20M1383525(FOCS20-175-FOCS20-204)Online publication date: 18-Apr-2022
https://doi.org/10.1137/20M1383525
Chen LKyng RLiu YPeng RGutenberg MSachdeva S(2022)Maximum Flow and Minimum-Cost Flow in Almost-Linear Time2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00064(612-623)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00064
Bernstein AGutenberg MSaranurak T(2022)Deterministic Decremental SSSP and Approximate Min-Cost Flow in Almost-Linear Time2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00100(1000-1008)Online publication date: Feb-2022
https://doi.org/10.1109/FOCS52979.2021.00100
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
Axiotis KMadry AVladu A(2022)Faster Sparse Minimum Cost Flow by Electrical Flow Localization2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00059(528-539)Online publication date: Feb-2022
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