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 NP-complete problems. Resolving the parameterized complexity status of DFVS was a long-standing open problem until Chen et al. (2008...
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| Veröffentlicht in: | Discrete optimization Jg. 46; S. 100740 |
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| Abstract | 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. (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 (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. |
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| AbstractList | 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. (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 (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. |
| ArticleNumber | 100740 |
| Author | Marx, Dániel Göke, Alexander Mnich, Matthias |
| Author_xml | – sequence: 1 givenname: Alexander surname: Göke fullname: Göke, Alexander email: alexander.goeke@tuhh.de organization: Hamburg University of Technology, Institute for Algorithms and Complexity, Hamburg, Germany – sequence: 2 givenname: Dániel surname: Marx fullname: Marx, Dániel email: dmarx@mpi-inf.mpg.de organization: CISPA Helmholtz Center for Information Security, Saarbrücken, Germany – sequence: 3 givenname: Matthias orcidid: 0000-0002-4721-5354 surname: Mnich fullname: Mnich, Matthias email: matthias.mnich@tuhh.de organization: Hamburg University of Technology, Institute for Algorithms and Complexity, Hamburg, Germany |
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| Cites_doi | 10.1137/12086217X 10.1002/jgt.21894 10.4086/toc.2013.v009a024 10.1007/s00453-012-9667-x 10.1145/2700209 10.1007/BF01200760 10.1137/090756144 10.1145/347476.347478 10.1137/1.9781611975031.125 10.1016/j.jcss.2017.04.004 10.1016/0304-3975(80)90009-2 10.1137/1.9781611975994.134 10.1007/PL00009191 10.4086/toc.2016.v012a006 10.1145/1411509.1411511 10.1007/s00453-016-0127-x |
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| Title | Parameterized algorithms for generalizations of Directed Feedback Vertex Set |
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