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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| Published in: | Discrete optimization Vol. 46; p. 100740 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Elsevier B.V
01.11.2022
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| Subjects: | |
| ISSN: | 1572-5286, 1873-636X |
| Online Access: | Get full text |
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| Summary: | 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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| ISSN: | 1572-5286 1873-636X |
| DOI: | 10.1016/j.disopt.2022.100740 |