  In the Directed Long Cycle Hitting Set} problem we are given a directed graph
$G$, and the task is to find a set $S$ of at most $k$ vertices/arcs such that
$G-S$ has no cycle of length longer than $\ell$. We show that the problem can
be solved in time $2^{\mathcal O(\ell k^3\log k + k^5\log k\log\ell)}\cdot
n^{\mathcal O(1)}$, that is, it is fixed-parameter tractable (FPT)
parameterized by $k$ and $\ell$. This algorithm can be seen as a far-reaching
generalization of the fixed-parameter tractability of {\sc Mixed Graph Feedback
Vertex Set} [Bonsma and Lokshtanov WADS 2011], which is already a common
generalization of the fixed-parameter tractability of (undirected) {\sc
Feedback Vertex Set} and the {\sc Directed Feedback Vertex Set} problems, two
classic results in parameterized algorithms. The algorithm requires significant
insights into the structure of graphs without directed cycles length longer
than $\ell$ and can be seen as an exact version of the approximation algorithm
following from the Erd{\H{o}}s-P{\'o}sa property for long cycles in directed
graphs proved by Kreutzer and Kawarabayashi [STOC 2015].

Data Structures and Algorithms