  Given two permutations $\sigma$ and $\pi$, the \textsc{Permutation Pattern}
problem asks if $\sigma$ is a subpattern of $\pi$. We show that the problem can
be solved in time $2^{O(\ell^2\log \ell)}\cdot n$, where $\ell=|\sigma|$ and
$n=|\pi|$. In other words, the problem is fixed-parameter tractable
parameterized by the size of the subpattern to be found.
  We introduce a novel type of decompositions for permutations and a
corresponding width measure. We present a linear-time algorithm that either
finds $\sigma$ as a subpattern of $\pi$, or finds a decomposition of $\pi$
whose width is bounded by a function of $|\sigma|$. Then we show how to solve
the \textsc{Permutation Pattern} problem in linear time if a bounded-width
decomposition is given in the input.
