Subgraph Sparsification and Nearly Optimal Ultrasparsifiers
We consider a variation of the spectral sparsification problem where we are required to keep a subgraph of the original graph. Formally, given a union of two weighted graphs \(G\) and \(W\) and an integer \(k\), we are asked to find a \(k\)-edge weighted graph \(W_k\) such that \(G+W_k\) is a good s...
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description | We consider a variation of the spectral sparsification problem where we are required to keep a subgraph of the original graph. Formally, given a union of two weighted graphs \(G\) and \(W\) and an integer \(k\), we are asked to find a \(k\)-edge weighted graph \(W_k\) such that \(G+W_k\) is a good spectral sparsifer of \(G+W\). We will refer to this problem as the subgraph (spectral) sparsification. We present a nontrivial condition on \(G\) and \(W\) such that a good sparsifier exists and give a polynomial time algorithm to find the sparsifer. %\(O(\frac{n}{k})\log n \tilde{O}(\log \log n)\) As a significant application of our technique, we show that for each positive integer \(k\), every \(n\)-vertex weighted graph has an \((n-1+k)\)-edge spectral sparsifier with relative condition number at most \(\frac{n}{k} \log n \tilde{O}(\log\log n)\) where \(\tilde{O}()\) hides lower order terms. Our bound is within a factor of \(\tilde{O}(\log \log n)\) from optimal. This nearly settles a question left open by Spielman and Teng about ultrasparsifiers, which is a key component in their nearly linear-time algorithms for solving diagonally dominant symmetric linear systems. We also present another application of our technique to spectral optimization in which the goal is to maximize the algebraic connectivity of a graph (e.g. turn it into an expander) with a limited number of edges. |
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This nearly settles a question left open by Spielman and Teng about ultrasparsifiers, which is a key component in their nearly linear-time algorithms for solving diagonally dominant symmetric linear systems. We also present another application of our technique to spectral optimization in which the goal is to maximize the algebraic connectivity of a graph (e.g. turn it into an expander) with a limited number of edges.</description><identifier>EISSN: 2331-8422</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Algorithms ; Graph theory ; Linear systems ; Mathematical analysis ; Optimization ; Polynomials ; Spectra</subject><ispartof>arXiv.org, 2009-12</ispartof><rights>2009. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). 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subjects | Algorithms Graph theory Linear systems Mathematical analysis Optimization Polynomials Spectra |
title | Subgraph Sparsification and Nearly Optimal Ultrasparsifiers |
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