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 diagonallydominant 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. * This paper is the second in a sequence of three papers expanding on material that appeared first under the title "Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems" [ST04]. The first paper, "A Local Clustering Algorithm for Massive Graphs and its Application to Nearly-Linear Time Graph Partitioning" [ST08a] contains graph partitioning algorithms that are used to construct the sparsifiers in this paper. The third paper, "Nearly-Linear Time Algorithms for Preconditioning and Solving Symmetric, Diagonally Dominant Linear Systems" [ST08b] contains the results on solving linear equations and approximating eigenvalues and eigenvectors.
Spielman and Teng introduced graph sparsification using spectral similarity of graph Laplacians REF .
Spectral Sparsification of Graphs
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