Probabilistic construction of deterministic algorithms: approximating packing integer programs
Journal of Computer and System Sciences - 27th IEEE Conference on Foundations of Computer Science October 27-29, 1986
Approximate counting, uniform generation and rapidly mixing Markov chains
Information and Computation
There are planar graphs almost as good as the complete graph
Journal of Computer and System Sciences
Approximating s-t minimum cuts in Õ(n2) time
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Balls and bins: a study in negative dependence
Random Structures & Algorithms
Efficiency of a Good But Not Linear Set Union Algorithm
Journal of the ACM (JACM)
Fast computation of low rank matrix approximations
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Support Theory for Preconditioning
SIAM Journal on Matrix Analysis and Applications
On clusterings: Good, bad and spectral
Journal of the ACM (JACM)
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Approximation Algorithms for Unique Games
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
SIAM Journal on Matrix Analysis and Applications
Local Graph Partitioning using PageRank Vectors
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Graph sparsification by effective resistances
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Spectral norm of random matrices
Combinatorica
Finding sparse cuts locally using evolving sets
Proceedings of the forty-first annual ACM symposium on Theory of computing
Proceedings of the forty-first annual ACM symposium on Theory of computing
Graph Sparsification by Effective Resistances
SIAM Journal on Computing
Spectral sparsification of graphs: theory and algorithms
Communications of the ACM
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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.