Spectral Sparsification of Graphs

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 logc 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. This material is based upon work supported by the National Science Foundation under Grant Nos. 0325630, 0324914, 0634957, 0635102 and 0707522. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Shang-Hua Teng wrote part of this paper while at MSR-NE lab and Boston University.