Approaching Optimality for Solving SDD Linear Systems

TL;DR: An algorithm that on input of an $n\times n$ symmetric diagonally dominant matrix A with $m$ non-zero entries and a vector $b$, computes a vector ${x} satisfying $| |{x}-A^{+}b| |_A|.

Abstract: We present an algorithm that on input of an $n$-vertex $m$-edge weighted graph $G$ and a value $k$, produces an {\em incremental sparsifier} $\hat{G}$ with $n-1 + m/k$ edges, such that the condition number of $G$ with $\hat{G}$ is bounded above by $\tilde{O}(k\log^2 n) $, with probability $1-p$. The algorithm runs in time $$\tilde{O}((m \log{n} + n\log^2{n})\log(1/p)).$$ As a result, we obtain an algorithm that on input of an $n\times n$ symmetric diagonally dominant matrix $A$ with $m$ non-zero entries and a vector $b$, computes a vector ${x}$ satisfying $| |{x}-A^{+}b| |_A