Parameterized Algorithms for Generalizations of Directed Feedback Vertex Set

The Directed Feedback Vertex Set (DFVS) problem takes as input a directed graph~$G$ and seeks a smallest vertex set~$S$ that hits all cycles in $G$. This is one of Karp’s 21 $\mathsf{N}\mathsf{P}$-complete problems. Resolving the parameterized complexity status of DFVS was a long-standing open problem until Chen et al. [STOC 2008, J. ACM 2008] showed its fixed-parameter tractability via a $4^{k}k!n^{\mathcal{O}(1)}$-time algorithm, where $k=\mathrm{\mid }S\mathrm{\mid }$. Here we show fixed-parameter tractability of two generalizations of DFVS:

• Find a smallest vertex set $S$ such that every strong component of $G-S$ has size at most~$s$: we give an algorithm solving this problem in time $4^k(ks+k+s)!\cdot n^{\mathcal{O}(1)}$. This generalizes an algorithm by Xiao [JCSS 2017] for the undirected version of the problem.
• Find a smallest vertex set $S$ such that every non-trivial strong component of $G-S$ is 1-out-regular: we give an algorithm solving this problem in time $2^{\mathcal{O}(k^3)}\cdot n^{\mathcal{O}(1)}$. We also solve the corresponding arc versions of these problems by fixed-parameter algorithms.