# Approximating the Exponential, the Lanczos Method and an \tilde{O}(m)-Time Spectral Algorithm for Balanced Separator

We give a novel spectral approximation algorithm for the balanced separator problem that, given a graph G, a constant balance b \in (0,1/2], and a parameter \gamma, either finds an \Omega(b)-balanced cut of conductance O(\sqrt{\gamma}) in G, or outputs a certificate that all b-balanced cuts in G have conductance at least \gamma, and runs in time \tilde{O}(m). This settles the question of designing asymptotically optimal spectral algorithms for balanced separator. Our algorithm relies on a variant of the heat kernel random walk and requires, as a subroutine, an algorithm to compute \exp(-L)v where L is the Laplacian of a graph related to G and v is a vector. Algorithms for computing the matrix-exponential-vector product efficiently comprise our next set of results. Our main result here is a new algorithm which computes a good approximation to \exp(-A)v for a class of symmetric positive semidefinite (PSD) matrices A and a given vector u, in time roughly \tilde{O}(m_A), where m_A is the number of non-zero entries of A. This uses, in a non-trivial way, the breakthrough result of Spielman and Teng on inverting symmetric and diagonally-dominant matrices in \tilde{O}(m_A) time. Finally, we prove e^{-x} can be uniformly approximated up to a small additive error, in a non-negative interval [a,b] with a polynomial of degree roughly \sqrt{b-a}. While this result is of independent interest in approximation theory, we show that, via the Lanczos method from numerical analysis, it yields a simple algorithm to compute \exp(-A)v for symmetric PSD matrices that runs in time roughly O(t_A \sqrt{||A||}), where t_A is time required for the computation of the vector Aw for given vector w. As an application, we obtain a simple and practical algorithm, with output conductance O(\sqrt{\gamma}), for balanced separator that runs in time \tilde{O}(m/\sqrt{\gamma}). This latter algorithm matches the running time, but improves on the approximation guarantee of the Evolving-Sets-based algorithm by Andersen and Peres for balanced separator.