  The Grid Theorem of Robertson and Seymour [JCTB, 1986], is one of the most
important tools in the field of structural graph theory, finding numerous
applications in the design of algorithms for undirected graphs. An analogous
version of the Grid Theorem in digraphs was conjectured by Johnson et al.
[JCTB, 2001], and proved by Kawarabayashi and Kreutzer [STOC, 2015]. Namely,
they showed that there is a function $f(k)$ such that every digraph of directed
tree-width at least $f(k)$ contains a cylindrical grid of size $k$ as a
butterfly minor and stated that their proof can be turned into an XP algorithm,
with parameter $k$, that either constructs a decomposition of the appropriate
width, or finds the claimed large cylindrical grid as a butterfly minor. In
this paper, we adapt some of the steps of the proof of Kawarabayashi and
Kreutzer to improve this XP algorithm into an FPT algorithm. Towards this, our
main technical contributions are two FPT algorithms with parameter $k$. The
first one either produces an arboreal decomposition of width $3k-2$ or finds a
haven of order $k$ in a digraph $D$, improving on the original result for
arboreal decompositions by Johnson et al. The second algorithm finds a
well-linked set of order $k$ in a digraph $D$ of large directed tree-width. As
tools to prove these results, we show how to solve a generalized version of the
problem of finding balanced separators for a given set of vertices $T$ in FPT
time with parameter $|T|$, a result that we consider to be of its own interest.
