# A fixed-parameter algorithm for the directed feedback vertex set problem

@article{Chen2008AFA, title={A fixed-parameter algorithm for the directed feedback vertex set problem}, author={Jianer Chen and Yang Liu and Songjian Lu and Barry O’Sullivan and Igor Razgon}, journal={J. ACM}, year={2008}, volume={55}, pages={21:1-21:19} }

The (parameterized) FEEDBACK VERTEX SET problem on directed graphs (i.e., the DFVS problem) is defined as follows: given a directed graph <i>G</i> and a parameter <i>k</i>, either construct a feedback vertex set of at most <i>k</i> vertices in <i>G</i> or report that no such a set exists. It has been a well-known open problem in parameterized computation and complexity whether the DFVS problem is fixed-parameter tractable, that is, whether the problem can be solved in time <i>f</i>(<i>k</i>)<i…

## 264 Citations

### Directed Subset Feedback Vertex Set Is Fixed-Parameter Tractable

- Computer ScienceTALG
- 2015

The random sampling of important separators technique is reformulated in an abstract way that can be used with a general family of transversal problems and will be useful for showing the FPT of other problems in directed graphs.

### A 4k2 kernel for feedback vertex set

- Mathematics, Computer ScienceTALG
- 2010

It is proved that given an undirected graph G, one can compute, in polynomial time in <i>n</i>, a graph G with at most 4-k vertices and an integer such that G has a feedback vertex set of size at most k.

### Tight bounds and a fast FPT algorithm for directed Max-Leaf Spanning Tree

- Computer ScienceTALG
- 2011

To obtain the complexity bound in the case of out-branchings, it is proved that when all arcs of <i>D</i> are part of at least one out- Branching, ℓ<sub><i>s</i></sub> (<i-i>) ≥ℓ(<i-d</i))/3.

### A Linear Time Parameterized Algorithm for Directed Feedback Vertex Set

- Computer Science, MathematicsArXiv
- 2016

The algorithm given is the first algorithm for DFVS with linear dependence on input size and the asymptotic dependence of the running time of the algorithm matches up to a factor of $k$ the algorithm of Chen, Liu, Lu, O'Sullivan and Razgon.

### Fixed-parameter tractability of multicut parameterized by the size of the cutset

- Computer ScienceSTOC '11
- 2011

It is unlikely that an algorithm with running time of the form f(p) ⋅ n<sup>O(1)</sup> exists for the directed version of the problem, as it is shown to be W[1]-hard parameterized by the size of the cutset.

### Towards a Polynomial Kernel for Directed Feedback Vertex Set

- Mathematics, Computer ScienceMFCS
- 2017

Two main contributions are provided: a polynomial kernel for this problem on general instances, and a linear kernel for the case where the input digraph is embeddable on a surface of bounded genus.

### Slightly superexponential parameterized problems

- Computer ScienceSODA '11
- 2011

It is shown that the dependence on k in the running time of the best known algorithms cannot be improved to single exponential and three natural problems, arising from three different domains are proved to be solvable in time.

### Finding small separators in linear time via treewidth reduction

- MathematicsTALG
- 2013

A method for reducing the treewidth of a graph while preserving all of its minimal separators up to a certain fixed size is presented, and this technique turns out to be relevant for H-coloring problems as well as cardinality constrained variants of the classical H- Coloring problem.

### On the Parameterized Complexity of Deletion to H-free Strong Components

- Mathematics, Computer ScienceMFCS
- 2020

The main result is a proof that this problem is fixed-parameter tractable parameterized by the size of the deletion set if H only contains rooted graphs or if H contains at least one directed path.

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