%0 Journal Article %1 DBLP:journals/corr/abs-0808-4134 %A Spielman, Daniel A. %A Teng, Shang-Hua %D 2008 %J CoRR %K eigenvalues graph.theory sparsification spectral.graph.theory %T Spectral Sparsification of Graphs %U http://arxiv.org/abs/0808.4134 %V abs/0808.4134 %X We introduce a new notion of graph sparsificaiton 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 $\softOm$, where $m$ is the number of edges in the original graph. This construction is a key component of a nearly-linear time algorithm for solving linear equations in diagonally-dominant matrcies. 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.

@article{DBLP:journals/corr/abs-0808-4134, abstract = {We introduce a new notion of graph sparsificaiton 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 $\softO{m}$, where $m$ is the number of edges in the original graph. This construction is a key component of a nearly-linear time algorithm for solving linear equations in diagonally-dominant matrcies. 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. }, added-at = {2011-03-29T00:38:55.000+0200}, author = {Spielman, Daniel A. and Teng, Shang-Hua}, bibsource = {DBLP, http://dblp.uni-trier.de}, biburl = {https://www.bibsonomy.org/bibtex/23fdd841acda82fa789193203447a03d6/ytyoun}, interhash = {b141d98b33dc29fdd418232c8b3a4c4e}, intrahash = {3fdd841acda82fa789193203447a03d6}, journal = {CoRR}, keywords = {eigenvalues graph.theory sparsification spectral.graph.theory}, timestamp = {2016-11-16T07:55:02.000+0100}, title = {Spectral Sparsification of Graphs}, url = {http://arxiv.org/abs/0808.4134}, volume = {abs/0808.4134}, year = 2008 }