  Given a graph $G$ and an integer $k$, the Feedback Vertex Set (FVS) problem
asks if there is a vertex set $T$ of size at most $k$ that hits all cycles in
the graph. The fixed-parameter tractability status of FVS in directed graphs
was a long-standing open problem until Chen et al. (STOC '08) showed that it is
FPT by giving a $4^{k}k!n^{O(1)}$ time algorithm. In the subset versions of
this problems, we are given an additional subset $S$ of vertices (resp., edges)
and we want to hit all cycles passing through a vertex of $S$ (resp. an edge of
$S$). Recently, the Subset Feedback Vertex Set in undirected graphs was shown
to be FPT by Cygan et al. (ICALP '11) and independently by Kakimura et al.
(SODA '12). We generalize the result of Chen et al. (STOC '08) by showing that
Subset Feedback Vertex Set in directed graphs can be solved in time
$2^{O(k^3)}n^{O(1)}$. By our result, we complete the picture for feedback
vertex set problems and their subset versions in undirected and directed
graphs. Besides proving the fixed-parameter tractability of Directed Subset
Feedback Vertex Set, we reformulate the random sampling of important separators
technique in an abstract way that can be used for a general family of
transversal problems. Moreover, we modify the probability distribution used in
the technique to achieve better running time; in particular, this gives an
improvement from $2^{2^{O(k)}}$ to $2^{O(k^2)}$ in the parameter dependence of
the Directed Multiway Cut algorithm of Chitnis et al. (SODA '12).
