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 $\mathrm{NP}$-complete problems. Resolving the parameterized complexity status of DFVS was a long-standing open problem until Chen et al. (2008) showed its fixed-parameter tractability via a $4^{k}k!n^{O\left(1\right)}$-time algorithm, where $k=\left|S\right|$.

Here we show fixed-parameter tractability of two generalizations of DFVS:

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  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 $4^k(ks+k+s)!\cdot n^{O\left(1\right)}$. This generalizes an algorithm by Xiao (2017) for the undirected version of the problem.

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  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 $2^{O\left(k^3\right)}\cdot n^{O\left(1\right)}$.

We also solve the corresponding arc versions of these problems by fixed-parameter algorithms.