0 Approaching optimality for solving SDD linear systems ∗

We present an algorithm that on input a graph G with n vertices and m+ n− 1 edges and a value k, produces an incremental sparsifier Ĝ with n − 1 +m/k edges, such that the condition number of G with Ĝ is bounded above by Õ(k log n), with probability 1− p. The algorithm runs in time Õ((m log n+ n log n) log(1/p)). As a result, we obtain an algorithm that on input an n × n symmetric diagonally dominant matrix A with m+n− 1 non-zero entries and a vector b, computes a vector x̄ satisfying ||x̄−Ab||A < !||Ab||A, in time Õ(m log n log(1/!)). The solver is based on a recursive application of the incremental sparsifier that produces a hierarchy of graphs which is then used to construct a recursive preconditioned Chebyshev iteration.