A ug 2 01 0 Approaching optimality for solving SDD linear systems ∗

We present an algorithm that on input of an $n$-vertex $m$-edge weighted graph $G$ and a value $k$ produces an incremental sparsifier $\hat{G}$ with $n-1 + m/k$ edges, such that the relative condition number of $G$ with $\hat{G}$ is bounded above by $\tilde{O}(k\log^2 n)$, with probability $1-p$ (we use the $\tilde{O}()$ notation to hide a factor of at most $(\log\log n)^4$). 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 $m$ nonzero entries and a vector $b$ computes a vector ${x}$ satisfying $||{x}-A^{+}b||_A<\epsilon ||A^{+}b||_A $, in expected time $\tilde{O}(m\log^2{n}\log(1/\epsilon)).$ The solver is based on repeated applications of the incremental sparsifier that produces a chain of graphs which is then used as input to the recursive preconditioned Chebyshev iteration.