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

The Mixing Time of the Dikin Walk in a Polytope - A Simple Proof

We study the mixing time of the Dikin walk in a polytope - a random walk based on the log-barrier from the interior point method literature. This walk, and a close variant, were studied by Narayanan (2016) and Kannan-Narayanan (2012). Bounds on its mixing time are important for algorithms for sampling and optimization over polytopes. Here, we provide a simple proof of their result that this random walk mixes in time O(mn) for an n-dimensional polytope described using m inequalities.Comment: 5 pages, published in Operations Research Letter

A New Approach to Estimating Effective Resistances and Counting Spanning Trees in Expander Graphs

We demonstrate that for expander graphs, for all $\epsilon > 0,$ there exists a data structure of size $\widetilde{O}(n\epsilon^{-1})$ which can be used to return $(1 + \epsilon)$-approximations to effective resistances in $\widetilde{O}(1)$ time per query. Short of storing all effective resistances, previous best approaches could achieve $\widetilde{O}(n\epsilon^{-2})$ size and $\widetilde{O}(\epsilon^{-2})$ time per query by storing Johnson-Lindenstrauss vectors for each vertex, or $\widetilde{O}(n\epsilon^{-1})$ size and $\widetilde{O}(n\epsilon^{-1})$ time per query by storing a spectral sketch. Our construction is based on two key ideas: 1) $\epsilon^{-1}$-sparse, $\epsilon$-additive approximations to $DL^+1_u$ for all $u,$ can be used to recover $(1 + \epsilon)$-approximations to the effective resistances, 2) In expander graphs, only $\widetilde{O}(\epsilon^{-1})$ coordinates of a vector similar to $DL^+1_u$ are larger than $\epsilon.$ We give an efficient construction for such a data structure in $\widetilde{O}(m + n\epsilon^{-2})$ time via random walks. This results in an algorithm for computing $(1+\epsilon)$-approximate effective resistances for $s$ vertex pairs in expanders that runs in $\widetilde{O}(m + n\epsilon^{-2} + s)$ time, improving over the previously best known running time of $m^{1 + o(1)} + (n + s)n^{o(1)}\epsilon^{-1.5}$ for $s = \omega(n\epsilon^{-0.5}).$ We employ the above algorithm to compute a $(1+\delta)$-approximation to the number of spanning trees in an expander graph, or equivalently, approximating the (pseudo)determinant of its Laplacian in $\widetilde{O}(m + n^{1.5}\delta^{-1})$ time. This improves on the previously best known result of $m^{1+o(1)} + n^{1.875+o(1)}\delta^{-1.75}$ time, and matches the best known size of determinant sparsifiers