N ov 2 00 3 Nearly - Linear Time Algorithms for Graph Partitioning , Graph Sparsification , and Solving Linear Systems third draft

We develop nearly-linear time algorithms for approximately solving sparse symmetric diagonally-dominant linear systems. In particular, for every β > 0 we present a linearsystem solver that, given an n-by-n symmetric diagonally-dominant matrix A with m nonzero entries and an n-vector b, produces a vector x̃ within relative distance ǫ of the solution to Ax = b in time m ( log + log(nκ(A)/ǫ) ) +min ( m,n logκ(A) ) log(nκ(A)/ǫ)2 √ log n log log , where κf (A) is the log of the ratio of the largest to smallest non-zero eigenvalue of A. We note that log(κf (A)) = O(L log n), where L is the logarithm of the ratio of the largest to smallest non-zero entry of A. We remark that while our algorithm is designed for sparse matrices, even for dense matrices the dominant term in its complexity is O(n). Our algorithm exploits two novel tools. The first is a nearly-linear time algorithm for approximately computing crude graph partitions. For any graph G having a cut of sparsity φ and balance b, this algorithm outputs a cut of sparsity at most O(φ log n) and balance b(1− ǫ) in time m((logm)/φ). Using this graph partitioning algorithm, we design fast graph sparsifiers and graph ultrasparsifiers. On input a weighted graph G with Laplacian matrix L and an ǫ > 0, the graph sparsifier produces a weighted graph G̃ with Laplacian matrix L̃ such that G̃ has n(log n)/ǫ edges and such that for all x ∈ IR, x T L̃x ≤ xLx ≤ (1 + ǫ)x L̃x . The ultra-sparsifier takes as input a parameter t and outputs a graph G̃ with (n−1)+ tn edges such that for all x ∈ IR x T L̃x ≤ xLx ≤ (n/t)x L̃x . Both algorithms run in time m log m. These ultra-sparsifiers almost asymptotically optimize the potential of the combinatorial preconditioners introduced by Vaidya. [email protected] † [email protected]