Parameterized Algorithms for Generalizations of Directed Feedback Vertex Set

  • Alexander GökeEmail author
  • Dániel Marx
  • Matthias Mnich
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11485)


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 \(\mathsf {NP}\)-complete problems. Resolving the parameterized complexity status of DFVS was a long-standing open problem until Chen et al. in 2008 showed its fixed-parameter tractability via a \(4^kk! n^{\mathcal {O}(1)}\)-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 \(4^k(ks+k+s)!\cdot n^{\mathcal {O}(1)}\).

  • 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.


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Authors and Affiliations

  1. 1.Universität BonnBonnGermany
  2. 2.SZTAKIBudapestHungary

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