Abstract

We introduce a new notion of graph sparsification based on spectral similarity of graph Laplacians: spectral sparsification requires that the Laplacian quadratic form of the sparsifier approximate that of the original. This is equivalent to saying that the Laplacian of the sparsifier is a good preconditioner for the Laplacian of the original. We prove that every graph has a spectral sparsifier of nearly linear size. Moreover, we present an algorithm that produces spectral sparsifiers in time $O(m\log^{c}m)$, where m is the number of edges in the original graph and c is some absolute constant. This construction is a key component of a nearly linear time algorithm for solving linear equations in diagonally dominant matrices. Our sparsification algorithm makes use of a nearly linear time algorithm for graph partitioning that satisfies a strong guarantee: if the partition it outputs is very unbalanced, then the larger part is contained in a subgraph of high conductance.

MSC codes

  1. 68R10
  2. 05C50

Keywords

  1. graph Laplacian
  2. sparsification
  3. graph partitioning

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Published In

cover image SIAM Journal on Computing
SIAM Journal on Computing
Pages: 981 - 1025
ISSN (online): 1095-7111

History

Submitted: 29 December 2008
Accepted: 18 February 2011
Published online: 12 July 2011

MSC codes

  1. 68R10
  2. 05C50

Keywords

  1. graph Laplacian
  2. sparsification
  3. graph partitioning

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