Solving Sparse, Symmetric, Diagonally-Dominant Linear Systems in Time 0(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 of the solution to Ax = b in time O(m log(n f (A)= )), where f (A) is the log of the ratio of the largest to smallest non-zero eigenvalue of A. In particular, log( f (A)) = O(b log n), where b is the logarithm of the ratio of the largest to smallest non-zero entry of A. If the graph of A has genus m or does not have a Km minor, then the exponent of m can be improved to the minimum of 1 + 5 and (9=8)(1+ ). The key contribution of our work is an extension of Vaidya’s techniques for constructing and analyzing combinatorial preconditioners.