  The notions of bounded expansion and nowhere denseness have been applied very
successfully in algorithmic graph theory. We study the corresponding notions of
directed bounded expansion and nowhere crownfulness on directed graphs. We show
that many of the algorithmic tools that were developed for undirected bounded
expansion classes can, with some care, also be applied in their directed
counterparts, and thereby we highlight a rich algorithmic structure theory of
directed bounded expansion classes.
  More specifically, we show that the directed Steiner tree problem is
fixed-parameter tractable on any class of directed bounded expansion
parameterized by the number $k$ of non-terminals plus the maximal diameter $s$
of a strongly connected component in the subgraph induced by the terminals. Our
result strongly generalizes a result of Jones et al., who proved that the
problem is fixed parameter tractable on digraphs of bounded degeneracy if the
set of terminals is required to be acyclic.
  We furthermore prove that for every integer $r\geq 1$, the distance-$r$
dominating set problem can be approximated up to a factor $O(\log k)$ and the
connected distance-$r$ dominating set problem can be approximated up to a
factor $O(k\cdot \log k)$ on any class of directed bounded expansion, where $k$
denotes the size of an optimal solution. If furthermore, the class is nowhere
crownful, we are able to compute a polynomial kernel for distance-$r$
dominating sets. Polynomial kernels for this problem were not known to exist on
any other existing digraph measure for sparse classes.
