Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: q-alg
36
The categories of this paper are: cmp-lg
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The categories of this paper are: comp-gas
7
The categories of this paper are: hep-th
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The categories of this paper are: q-fin
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
3
The categories of this paper are: physics
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: astro-ph
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The categories of this paper are: stat
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The categories of this paper are: adap-org
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The categories of this paper are: cond-mat
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The categories of this paper are: eess