@online{Goeke_arXiv2003.05267,
TITLE = {Hitting Long Directed Cycles is Fixed-Parameter Tractable},
AUTHOR = {G{\"o}ke, Alexander and Marx, D{\'a}niel and Mnich, Matthias},
LANGUAGE = {eng},
URL = {https://arxiv.org/abs/2003.05267},
EPRINT = {2003.05267},
EPRINTTYPE = {arXiv},
YEAR = {2020},
MARGINALMARK = {$\bullet$},
ABSTRACT = {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].},
}