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 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 result of Spielman and Teng on inverting SDD 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 PSD matrices that runs in time roughly O(t_A \sqrt{||A||}), where t_A is the time required for 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 algorithm by Andersen and Peres. 
Lorenzo Orecchia, Sushant Sachdeva, Nisheeth K. Vishnoi
cs.DS cs.NA math.CA math.NA