43 research outputs found
An SDP-Based Algorithm for Linear-Sized Spectral Sparsification
For any undirected and weighted graph with vertices and
edges, we call a sparse subgraph of , with proper reweighting of the
edges, a -spectral sparsifier if holds for any , where and
are the respective Laplacian matrices of and . Noticing that
time is needed for any algorithm to construct a spectral sparsifier and a
spectral sparsifier of requires edges, a natural question is to
investigate, for any constant , if a -spectral
sparsifier of with edges can be constructed in time,
where the notation suppresses polylogarithmic factors. All previous
constructions on spectral sparsification require either super-linear number of
edges or time.
In this work we answer this question affirmatively by presenting an algorithm
that, for any undirected graph and , outputs a
-spectral sparsifier of with edges in
time. Our algorithm is based on three novel
techniques: (1) a new potential function which is much easier to compute yet
has similar guarantees as the potential functions used in previous references;
(2) an efficient reduction from a two-sided spectral sparsifier to a one-sided
spectral sparsifier; (3) constructing a one-sided spectral sparsifier by a
semi-definite program.Comment: To appear at STOC'1
Approximate Gaussian Elimination for Laplacians: Fast, Sparse, and Simple
We show how to perform sparse approximate Gaussian elimination for Laplacian
matrices. We present a simple, nearly linear time algorithm that approximates a
Laplacian by a matrix with a sparse Cholesky factorization, the version of
Gaussian elimination for symmetric matrices. This is the first nearly linear
time solver for Laplacian systems that is based purely on random sampling, and
does not use any graph theoretic constructions such as low-stretch trees,
sparsifiers, or expanders. The crux of our analysis is a novel concentration
bound for matrix martingales where the differences are sums of conditionally
independent variables