Solving sparse, symmetric, diagonally-dominant linear systems in time O(m1.31

We present a linear-system solver that, given an n-by-n symmetric positive semi-definite, diagonally dominant matrix A with m non-zero entries and an n-vector b, produces a vector x˜ within relative distance ε of the solution to Ax = b in time O(m^1.31log(n/ε)b^O(1)), where b is the log of the ratio of the largest to smallest non-zero entry of A. If the graph of A has genus m^2θ or does not have a K_mθ minor, then the exponent of m can be improved to the minimum of 1 + 5θ and (9/8)(1 + θ). The key contribution of our work is an extension of Vaidya´s techniques for constructing and analyzing combinatorial preconditioners.