NDLI: Twice-Ramanujan Sparsifiers (SMJCAT Vol. 41 Issue. 6)

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Batson, Joshua Spielman, Daniel A. Srivastava, Nikhil
Copyright Year	2012
Abstract	We prove that every graph has a spectral sparsifier with a number of edges linear in its number of vertices. As linear-sized spectral sparsifiers of complete graphs are expanders, our sparsifiers of arbitrary graphs can be viewed as generalizations of expander graphs. In particular, we prove that for every $d>1$ and every undirected, weighted graph $G=(V,E,w)$ on $n$ vertices, there exists a weighted graph $H=(V,F,\tilde{w})$ with at most $\lceil d(n-1)\rceil$ edges such that for every $x\in\mathbb{R}^{V}$, $x^{T}L_{G}x\leq x^{T}L_{H}x\leq\bigl(\frac{d+1+2\sqrt{d}}{d+1-2\sqrt{d}}\bigr)\cdot x^{T}L_{G}x$, where $L_{G}$ and $L_{H}$ are the Laplacian matrices of $G$ and $H$, respectively. Thus, $H$ approximates $G$ spectrally at least as well as a Ramanujan expander with $dn/2$ edges approximates the complete graph. We give an elementary deterministic polynomial time algorithm for constructing $H$.
Starting Page	1704
Ending Page	1721
Page Count	18
File Format	PDF
ISSN	00975397
DOI	10.1137/090772873
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	6 (Special Section on the Forty-First Annual ACM Symposium on Theory of Computing (STOC 2009))
Volume Number	41
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2012-12-18
Access Restriction	Subscribed
Subject Keyword	spectral sparsification paving conjecture barrier method graph sparsification rank-one updates Positive matrices and their generalizations; cones of matrices Graph theory spectral graph theory sparse quadratic forms
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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