We develop two different methods to achieve subexponential time parameterized
algorithms for problems on sparse directed graphs. We exemplify our approaches
with two well studied problems.
For the first problem, {\sc k-Leaf Out-Branching}, which is to find an
oriented spanning tree with at least k leaves, we obtain an algorithm solving
the problem in time 2O(klogk)n+nO(1) on directed graphs
whose underlying undirected graph excludes some fixed graph H as a minor. For
the special case when the input directed graph is planar, the running time can
be improved to 2O(k)n+nO(1). The second example is a
generalization of the {\sc Directed Hamiltonian Path} problem, namely {\sc
k-Internal Out-Branching}, which is to find an oriented spanning tree with at
least k internal vertices. We obtain an algorithm solving the problem in time
2O(klogk)+nO(1) on directed graphs whose underlying
undirected graph excludes some fixed apex graph H as a minor. Finally, we
observe that for any ϵ>0, the {\sc k-Directed Path} problem is
solvable in time O((1+ϵ)knf(ϵ)), where f is some
function of \ve.
Our methods are based on non-trivial combinations of obstruction theorems for
undirected graphs, kernelization, problem specific combinatorial structures and
a layering technique similar to the one employed by Baker to obtain PTAS for
planar graphs