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}
}
• Published 1 October 2008
• Computer Science
• J. ACM
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

Figures from this paper

Directed Subset Feedback Vertex Set Is Fixed-Parameter Tractable

• Computer Science
TALG
• 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

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 Science
TALG
• 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, Mathematics
ArXiv
• 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 Science
STOC '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 Science
MFCS
• 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 Science
SODA '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

• Mathematics
TALG
• 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 Science
MFCS
• 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.

References

SHOWING 1-10 OF 52 REFERENCES

Faster fixed parameter tractable algorithms for finding feedback vertex sets

• Computer Science, Mathematics
TALG
• 2006
A feedback vertex set of a graph is a set of vertices whose removal results in an acyclic graph and there is a cycle of length at most 6/ε (for ε ≥ 1/2, the authors can even improve this to just 6).

Approximating Minimum Feedback Sets and Multicuts in Directed Graphs

• Computer Science, Mathematics
Algorithmica
• 1998
A combinatorial algorithm that computes a (1+ɛ) approximation to the fractional optimal feedback vertex set, and a generalization of these problems, in which the feedback set has to intersect only a subset of the directed cycles in the graph.

A Cubic Kernel for Feedback Vertex Set

It is shown that the Feedback Vertex Set problem on unweighted, undirected graphs has a kernel of cubic size and can be used as a first step of an FPT algorithm for FEEDBACK VERTEX SET, but also as a preprocessing heuristic for the problem.

Parameterized Algorithms for Feedback Vertex Set

• Mathematics
IWPEC
• 2004
It is shown that for several special classes of graphs the feedback vertex set problem can be solved in time c k n O(1) for some constant c.

A 2-Approximation Algorithm for the Undirected Feedback Vertex Set Problem

• Mathematics, Computer Science
SIAM J. Discret. Math.
• 1999
A simple and efficient approximation algorithm with performance ratio of at most 2 is presented, improving previous best bounds for either weighted or unweighted cases of the vertex cover problem.

On Linear Time Minor Tests with Depth-First Search

If at least one graph H i is a minor of a 2 × k grid graph, and at leastOne graph Hi is aMinor of a circus graph, then one can test in $$\mathcal{O}$$(n) time whether a given graph G contains at leastone graph H∈{H1, ..., H c } as a minor.

• Mathematics