scholarly journals Towards a Polynomial Kernel for Directed Feedback Vertex Set

Algorithmica ◽  
2020 ◽  
Author(s):  
Benjamin Bergougnoux ◽  
Eduard Eiben ◽  
Robert Ganian ◽  
Sebastian Ordyniak ◽  
M. S. Ramanujan
Keyword(s):  
Undirected Graph ◽  
Polynomial Kernel ◽  
Directed Cycle ◽  
Feedback Vertex Set ◽  
Open Problems ◽  
Fixed Parameter Tractable ◽  
Parameterized Algorithmics ◽  
Fixed Parameter ◽  
Vertex Set ◽  
Directed Feedback Vertex Set

Abstract In the Directed Feedback Vertex Set (DFVS) problem, the input is a directed graph D and an integer k. The objective is to determine whether there exists a set of at most k vertices intersecting every directed cycle of D. DFVS was shown to be fixed-parameter tractable when parameterized by solution size by Chen et al. (J ACM 55(5):177–186, 2008); since then, the existence of a polynomial kernel for this problem has become one of the largest open problems in the area of parameterized algorithmics. Since this problem has remained open in spite of the best efforts of a number of prominent researchers and pioneers in the field, a natural step forward is to study the kernelization complexity of DFVS parameterized by a natural larger parameter. In this paper, we study DFVS parameterized by the feedback vertex set number of the underlying undirected graph. We provide two main contributions: 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.

scholarly journals Subset Feedback Vertex Set Is Fixed-Parameter Tractable

2013 ◽  
Vol 27 (1) ◽  
pp. 290-309 ◽  
Author(s):  
Marek Cygan ◽  
Marcin Pilipczuk ◽  
Michał Pilipczuk ◽  
Jakub Onufry Wojtaszczyk
Keyword(s):  
Feedback Vertex Set ◽  
Fixed Parameter Tractable ◽  
Fixed Parameter ◽  
Vertex Set

scholarly journals Subset Feedback Vertex Set Is Fixed-Parameter Tractable

2011 ◽  
pp. 449-461 ◽  
Author(s):  
Marek Cygan ◽  
Marcin Pilipczuk ◽  
Michał Pilipczuk ◽  
Jakub Onufry Wojtaszczyk
Keyword(s):  
Feedback Vertex Set ◽  
Fixed Parameter Tractable ◽  
Fixed Parameter ◽  
Vertex Set

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

2008 ◽  
Vol 55 (5) ◽  
pp. 1-19 ◽  
Author(s):  
Jianer Chen ◽  
Yang Liu ◽  
Songjian Lu ◽  
Barry O'sullivan ◽  
Igor Razgon
Keyword(s):  
Feedback Vertex Set ◽  
Fixed Parameter ◽  
Vertex Set ◽  
Feedback Vertex Set Problem ◽  
Fixed Parameter Algorithm ◽  
Directed Feedback Vertex Set

scholarly journals Directed Subset Feedback Vertex Set Is Fixed-Parameter Tractable

2015 ◽  
Vol 11 (4) ◽  
pp. 1-28 ◽  
Author(s):  
Rajesh Chitnis ◽  
Marek Cygan ◽  
Mohammataghi Hajiaghayi ◽  
Dániel Marx
Keyword(s):  
Feedback Vertex Set ◽  
Fixed Parameter Tractable ◽  
Fixed Parameter ◽  
Vertex Set

scholarly journals FPT Algorithms for Diverse Collections of Hitting Sets

Algorithms ◽  
2019 ◽  
Vol 12 (12) ◽  
pp. 254
Author(s):  
Julien Baste ◽  
Lars Jaffke ◽  
Tomáš Masařík ◽  
Geevarghese Philip ◽  
Günter Rote
Keyword(s):  
Network Flow ◽  
Hamming Distance ◽  
Side Information ◽  
Feedback Vertex Set ◽  
Fixed Parameter Tractable ◽  
Fpt Algorithms ◽  
Fixed Parameter ◽  
Vertex Set ◽  
Two Measures ◽  
Real World Problems

In this work, we study the d-Hitting Set and Feedback Vertex Set problems through the paradigm of finding diverse collections of r solutions of size at most k each, which has recently been introduced to the field of parameterized complexity. This paradigm is aimed at addressing the loss of important side information which typically occurs during the abstraction process that models real-world problems as computational problems. We use two measures for the diversity of such a collection: the sum of all pairwise Hamming distances, and the minimum pairwise Hamming distance. We show that both problems are fixed-parameter tractable in k + r for both diversity measures. A key ingredient in our algorithms is a (problem independent) network flow formulation that, given a set of ‘base’ solutions, computes a maximally diverse collection of solutions. We believe that this could be of independent interest.

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