## Abstract

In the Directed Feedback Vertex Set (DFVS) problem, we are given a digraph D on n vertices and a positive integer k, and the objective is to check whether there exists a set of vertices S such that F = D − S is an acyclic digraph. In a recent paper, Mnich and van Leeuwen [STACS 2016] studied the kernelization complexity of DFVS with an additional restriction on F—namely that F must be an out-forest, an out-tree, or a (directed) pumpkin—with an objective of shedding some light on the kernelization complexity of the DFVS problem, a well known open problem in the area. The vertex deletion problems corresponding to obtaining an out-forest, an out-tree, or a (directed) pumpkin are Out-forest/Out-tree/Pumpkin Vertex Deletion Set, respectively. They showed that Out-forest/Out-tree/Pumpkin Vertex Deletion Set admit polynomial kernels. Another open problem regarding DFVS is that, does DFVS admit an algorithm with running time 2 ^{O} ^{(} ^{k} ^{)}n^{O} ^{(} ^{1} ^{)}? We complement the kernelization programme of Mnich and van Leeuwen by designing fast FPT algorithms for the above mentioned problems. In particular, we design an algorithm for Out-forest Vertex Deletion Set that runs in time O(2.73 2 ^{k}n^{O} ^{(} ^{1} ^{)}) and algorithms for Pumpkin/Out-tree Vertex Deletion Set that runs in time O(2.56 2 ^{k}n^{O} ^{(} ^{1} ^{)}). As a corollary of our FPT algorithms and the recent result of Fomin et al. [STOC 2016] which gives a relation between FPT algorithms and exact algorithms, we get exact algorithms for Out-forest/Out-tree/Pumpkin Vertex Deletion Set that run in time O(1.63 3 ^{n}n^{O} ^{(} ^{1} ^{)}) , O(1.60 9 ^{n}n^{O} ^{(} ^{1} ^{)}) and O(1.60 9 ^{n}n^{O} ^{(} ^{1} ^{)}) , respectively.

Original language | American English |
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Pages (from-to) | 1880-1909 |

Number of pages | 30 |

Journal | Theory of Computing Systems |

Volume | 62 |

Issue number | 8 |

DOIs | |

State | Published - 1 Nov 2018 |

## Keywords

- Bounded search trees
- Fixed parameter tractability
- Out-forest
- Out-tree
- Pumpkin
- branching

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computational Theory and Mathematics