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 NP-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 4kk!nO(1)-time algorithm, where k=|S|. 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 4k(ks+k+s)!⋅nO(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 2O(k3)⋅nO(1). We also solve the corresponding arc versions of these problems by fixed-parameter algorithms.
2022-11
2024-06-28