research-article

Subgraph sparsification and nearly optimal ultrasparsifiers

Authors:

Alexandra Kolla,

Yury Makarychev,

Shang-Hua TengAuthors Info & Claims

STOC '10: Proceedings of the forty-second ACM symposium on Theory of computing

Pages 57 - 66

https://doi.org/10.1145/1806689.1806699

Published: 05 June 2010 Publication History

Abstract

We consider a variation of the spectral sparsification problem where we are required to keep a subgraph of the original graph. Formally, given a union of two weighted graphs G and W and an integer k, we are asked to find a k-edge weighted graph W_k such that G+W_k is a good spectral sparsifer of G+W. We will refer to this problem as the subgraph (spectral) sparsification. We present a nontrivial condition on G and W such that a good sparsifier exists and give a polynomial-time algorithm to find the sparsifer.

As an application of our technique, we show that for each positive integer k, every n-vertex weighted graph has an (n-1+k)-edge spectral sparsifier with relative condition number at most n/k log n, ~O(log log n) where ~O() hides lower order terms.

Our bound nearly settles a question left open by Spielman and Teng about ultrasparsifiers.

We also present another application of our technique to spectral optimization in which the goal is to maximize the algebraic connectivity of a graph (e.g. turn it into an expander) with a limited number of edges.

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Cited By

Elkin MShabat I(2023)Path-Reporting Distance Oracles with Logarithmic Stretch and Size O(n log log n)2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00141(2278-2311)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00141
Chan PLau LSchild AWong SZhou H(2022)Network Design for s-t Effective ResistanceACM Transactions on Algorithms10.1145/352258818:3(1-45)Online publication date: 11-Oct-2022
https://dl.acm.org/doi/10.1145/3522588
Lau LZhou H(2022)A Local Search Framework for Experimental DesignSIAM Journal on Computing10.1137/20M138654251:4(900-951)Online publication date: 1-Jan-2022
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Index Terms

Subgraph sparsification and nearly optimal ultrasparsifiers
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
2. Theory of computation
  1. Design and analysis of algorithms

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cover image ACM Conferences

STOC '10: Proceedings of the forty-second ACM symposium on Theory of computing

June 2010

812 pages

ISBN:9781450300506

DOI:10.1145/1806689

General Chair:
Michael Mitzenmacher
Harvard
,
Program Chair:
Leonard J. Schulman
Caltech

Copyright © 2010 ACM.

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Published: 05 June 2010

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June 5 - 8, 2010

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Cited By

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https://doi.org/10.1109/FOCS57990.2023.00141
Chan PLau LSchild AWong SZhou H(2022)Network Design for s-t Effective ResistanceACM Transactions on Algorithms10.1145/352258818:3(1-45)Online publication date: 11-Oct-2022
https://dl.acm.org/doi/10.1145/3522588
Lau LZhou H(2022)A Local Search Framework for Experimental DesignSIAM Journal on Computing10.1137/20M138654251:4(900-951)Online publication date: 1-Jan-2022
https://dl.acm.org/doi/10.1137/20M1386542
Lau LZhou H(2022)A Spectral Approach to Network DesignSIAM Journal on Computing10.1137/20M133076251:4(1018-1064)Online publication date: 1-Jan-2022
https://dl.acm.org/doi/10.1137/20M1330762
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
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Lau LZhou HMarx D(2021)A local search framework for experimental designProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458129(1039-1058)Online publication date: 10-Jan-2021
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Jambulapati ASidford AMarx D(2021)Ultrasparse ultrasparsifiers and faster laplacian system solversProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458097(540-559)Online publication date: 10-Jan-2021
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Elkin MMatar SMiller ACensor-Hillel K(2021)Ultra-Sparse Near-Additive EmulatorsProceedings of the 2021 ACM Symposium on Principles of Distributed Computing10.1145/3465084.3467926(235-246)Online publication date: 21-Jul-2021
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Zhang QGad ENouri BNa WNakhla M(2021)Simulation and Automated Modeling of Microwave Circuits: State-of-the-Art and Emerging TrendsIEEE Journal of Microwaves10.1109/JMW.2020.30337801:1(494-507)Online publication date: Jan-2021
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Lau LZhou HMakarychev KMakarychev YTulsiani MKamath GChuzhoy J(2020)A spectral approach to network designProceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3357713.3384313(826-839)Online publication date: 22-Jun-2020
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