Approaching optimality for solving SDD 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 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 logn+ n log n) log(1/p)). As a result, we obtain an algorithm that on input of an n × n symmetric diagonally dominant matrix A with m non-zero entries and a vector b, computes a vector x satisfying ||x − Ab||A < ǫ||Ab||A, in expected time Õ(m log n log(1/ǫ)). The solver is based on repeated applications of the incremental sparsifier that produces a chain of graphs which is then used as input to a recursive preconditioned Chebyshev iteration.