research-article

Sparse Sums of Positive Semidefinite Matrices

Authors:

Marcel K. De Carli Silva,

Nicholas J. A. Harvey,

Cristiane M. SatoAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 12, Issue 1

Article No.: 9, Pages 1 - 17

https://doi.org/10.1145/2746241

Published: 09 December 2015 Publication History

Abstract

Many fast graph algorithms begin by preprocessing the graph to improve its sparsity. A common form of this is spectral sparsification, which involves removing and reweighting the edges of the graph while approximately preserving its spectral properties. This task has a more general linear algebraic formulation in terms of approximating sums of rank-one matrices. This article considers a more general task of approximating sums of symmetric, positive semidefinite matrices of arbitrary rank. We present two deterministic, polynomial time algorithms for solving this problem. The first algorithm applies the pessimistic estimators of Wigderson and Xiao, and the second involves an extension of the method of Batson, Spielman, and Srivastava. These algorithms have several applications, including sparsifiers of hypergraphs, sparse solutions to semidefinite programs, sparsifiers of unique games, and graph sparsifiers with various auxiliary constraints.

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Cited By

Kılınç-Karzan FKüçükyavuz SLee DShafieezadeh-Abadeh S(2023)Technical Note—Conic Mixed-Binary Sets: Convex Hull Characterizations and ApplicationsOperations Research10.1287/opre.2020.0827Online publication date: 19-Jul-2023
https://doi.org/10.1287/opre.2020.0827
Hu XLin J(2023)Solving Graph Laplacians via Multilevel SparsifiersSIAM Journal on Scientific Computing10.1137/22M150393246:2(S378-S400)Online publication date: 30-Nov-2023
https://doi.org/10.1137/22M1503932
Apers Sde Wolf R(2022)Quantum Speedup for Graph Sparsification, Cut Approximation, and Laplacian SolvingSIAM Journal on Computing10.1137/21M139101851:6(1703-1742)Online publication date: 16-Dec-2022
https://doi.org/10.1137/21M1391018
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Index Terms

Sparse Sums of Positive Semidefinite Matrices
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Spectra of graphs
2. Theory of computation
  1. Design and analysis of algorithms
    1. Graph algorithms analysis
      1. Sparsification and spanners

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Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 12, Issue 1

Special Issue on SODA'12 and Regular Papers

February 2016

243 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/2846103

Editor:
Aravind Srinivasan
University of Maryland, USA

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Copyright © 2015 ACM.

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Publication History

Published: 09 December 2015

Accepted: 01 March 2015

Revised: 01 March 2014

Received: 01 October 2011

Published in TALG Volume 12, Issue 1

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https://doi.org/10.1287/opre.2020.0827
Hu XLin J(2023)Solving Graph Laplacians via Multilevel SparsifiersSIAM Journal on Scientific Computing10.1137/22M150393246:2(S378-S400)Online publication date: 30-Nov-2023
https://doi.org/10.1137/22M1503932
Apers Sde Wolf R(2022)Quantum Speedup for Graph Sparsification, Cut Approximation, and Laplacian SolvingSIAM Journal on Computing10.1137/21M139101851:6(1703-1742)Online publication date: 16-Dec-2022
https://doi.org/10.1137/21M1391018
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https://doi.org/10.1137/20M1330762
Ivanov GNaszódi MPolyanskii A(2020)Approximation of the average of some random matricesJournal of Functional Analysis10.1016/j.jfa.2020.108684279:7(108684)Online publication date: Oct-2020
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Liu YXu CZhan YLiu ZGuan JZhang H(2017)Incentive mechanism for computation offloading using edge computingComputer Networks: The International Journal of Computer and Telecommunications Networking10.1016/j.comnet.2017.03.015129:P2(399-409)Online publication date: 24-Dec-2017
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