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 ∈ (0, 1/2], and a parameter γ, either finds an Ω(b)balanced cut of conductance O( √ γ) in G, or outputs a certificate that all b-balanced cuts in G have conductance at least γ, and runs in time Õ(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 Õ(mA), where mA 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 Õ(mA) time. Finally, we prove that 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 √ 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(tA · √ ‖A‖), where tA 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( √ γ), for balanced separator that runs in time Õ(m/√γ). 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.