research-article

Expander flows, geometric embeddings and graph partitioning

Authors:

Umesh VaziraniAuthors Info & Claims

Journal of the ACM (JACM), Volume 56, Issue 2

Article No.: 5, Pages 1 - 37

https://doi.org/10.1145/1502793.1502794

Published: 17 April 2009 Publication History

Abstract

We give a O(√log n)-approximation algorithm for the sparsest cut, edge expansion, balanced separator, and graph conductance problems. This improves the O(log n)-approximation of Leighton and Rao (1988). We use a well-known semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in R^d, whose proof makes essential use of a phenomenon called measure concentration.

We also describe an interesting and natural “approximate certificate” for a graph's expansion, which involves embedding an n-node expander in it with appropriate dilation and congestion. We call this an expander flow.

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Chalermsook PKaul MMnich MSpoerhase JUniyal SVaz D(2024)Approximating Sparsest Cut in Low-treewidth Graphs via Combinatorial DiameterACM Transactions on Algorithms10.1145/363262320:1(1-20)Online publication date: 22-Jan-2024
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Index Terms

Expander flows, geometric embeddings and graph partitioning
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
2. Theory of computation
  1. Design and analysis of algorithms

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cover image Journal of the ACM

Journal of the ACM Volume 56, Issue 2

April 2009

190 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/1502793

Issue’s Table of Contents

Copyright © 2009 ACM.

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Publication History

Published: 17 April 2009

Accepted: 01 December 2008

Revised: 01 July 2008

Received: 01 April 2007

Published in JACM Volume 56, Issue 2

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Gu H(2024)Rethinking the implementation and application of the Benczur-Karger minimum cuts algorithmInternational Journal of ADVANCED AND APPLIED SCIENCES10.21833/ijaas.2024.07.00711:7(57-62)Online publication date: Jul-2024
https://doi.org/10.21833/ijaas.2024.07.007
Park MTabatabaee YRamavarapu VLiu BPailodi VRamachandran RKorobskiy DAyres FChacko GWarnow T(2024)Well-connectedness and community detectionPLOS Complex Systems10.1371/journal.pcsy.00000091:3(e0000009)Online publication date: 5-Nov-2024
https://doi.org/10.1371/journal.pcsy.0000009
Chalermsook PKaul MMnich MSpoerhase JUniyal SVaz D(2024)Approximating Sparsest Cut in Low-treewidth Graphs via Combinatorial DiameterACM Transactions on Algorithms10.1145/363262320:1(1-20)Online publication date: 22-Jan-2024
https://dl.acm.org/doi/10.1145/3632623
Kolmogorov VAgrawal KPetrank E(2024)A Simpler and Parallelizable O(√log n)-approximation Algorithm for Sparsest CutProceedings of the 36th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3626183.3659969(403-414)Online publication date: 17-Jun-2024
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