Sparsified Cholesky Solvers for SDD linear systems

We show that Laplacian and symmetric diagonally dominant (SDD) matrices can be well approximated by linear-sized sparse Cholesky factorizations. We show that these matrices have constant-factor approximations of the form L LT, where L is a lower-triangular matrix with a number of nonzero entries linear in its dimension. Furthermore linear systems in L and LT can be solved in O (n) work and O(lognlog2logn) depth, where n is the dimension of the matrix. We present nearly linear time algorithms that construct solvers that are almost this efficient. In doing so, we give the first nearly-linear work routine for constructing spectral vertex sparsifiers---that is, spectral approximations of Schur complements of Laplacian matrices.