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A Cubic Kernel for Feedback Vertex Set and Loop Cutset

Hans L. Bodlaender, Thomas C. van Dijk
2009 Theory of Computing Systems  
This kernelization algorithm can be used as the first step of an FPT algorithm for FEEDBACK VERTEX SET, but also as a preprocessing heuristic for FEEDBACK VERTEX SET.  ...  feedback vertex set of G.  ...  Acknowledgements We thank the anonymous referees of the paper for useful comments.  ... 
doi:10.1007/s00224-009-9234-2 fatcat:cwq2aws44bhmhd4r3qxbdqcvoq

A Cubic Kernel for Feedback Vertex Set [chapter]

Hans L. Bodlaender
STACS 2007  
This kernelization algorithm can be used as the first step of an FPT algorithm for Feedback Vertex Set, but also as a preprocessing heuristic for Feedback Vertex Set.  ...  Moreover, if G has a feedback vertex set of size at most k , then a minimum size feedback vertex set of G directly gives a minimum size feedback vertex set of size k .  ...  We claim that for each set S ⊆ V , S is a feedback vertex set in G, if and only if S is a feedback vertex set in G .  ... 
doi:10.1007/978-3-540-70918-3_28 dblp:conf/stacs/Bodlaender07 fatcat:blfcrzsppnfnrdn5fb2zo2cbbm

Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter

Bart M. P. Jansen, Hans L. Bodlaender, Marc Herbstritt
2011 Symposium on Theoretical Aspects of Computer Science  
be reduced to a polynomial in the size of the minimum FVS  In the language of parameterized complexity: Vertex Cover parameterized by the size of a feedback vertex set admits a cubic-The Weighted Vertex  ...  Vertex Cover using a "refined" parameter: Feedback Vertex Number We have studied data reduction for Vertex Cover using a "refined" parameter: Feedback Vertex Number  Kernel with O(|X| 3 ) vertices  Usage  ...  degree of the polynomial: cubic to quadratic?  ... 
doi:10.4230/lipics.stacs.2011.177 dblp:conf/stacs/JansenB11 fatcat:v6mfndllc5bsxctpwhranx52wa

A quadratic kernel for feedback vertex set [chapter]

Stéphan Thomassé
2009 Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms  
set of size at most k iff G has a feedback vertex set of size at most k .  ...  This result improves a previous O(k 11 ) kernel of Burrage et al. [6] , and a more recent cubic kernel of Bodlaender [3]. This problem was communicated by Fellows in [5] .  ...  We provide in this paper a quadratic kernel for FEEDBACK VERTEX SET which reduces the input graph G to a graph G with size at most 5k 2 +k.  ... 
doi:10.1137/1.9781611973068.13 fatcat:pnzy5eafojdsjcmu3swyzg4fym

A 4k2kernel for feedback vertex set

Stéphan Thomassé
2010 ACM Transactions on Algorithms  
set of size at most k iff G has a feedback vertex set of size at most k .  ...  This result improves a previous O(k 11 ) kernel of Burrage et al. [5] , and a more recent cubic kernel of Bodlaender [3] . This problem was communicated by Fellows in [4] .  ...  We provide in this paper a quadratic kernel for FEEDBACK VERTEX SET which reduces the input graph G to a graph G with size at most 4k 2 .  ... 
doi:10.1145/1721837.1721848 fatcat:raq6svwq6ncehczv6onenivzly

Safe Approximation and Its Relation to Kernelization [chapter]

Jiong Guo, Iyad Kanj, Stefan Kratsch
2012 Lecture Notes in Computer Science  
We introduce a notion of approximation, called safe approximation, for minimization problems that are subset problems.  ...  We demonstrate how the notion of safe approximation can be useful in designing kernelization algorithms for certain fixed-parameter tractable problems.  ...  The authors would like to thank Bart Jansen for suggesting the nice example used in the proof of Theorem 4.  ... 
doi:10.1007/978-3-642-28050-4_14 fatcat:2c7xqcqavrh7jae74ms2ndbowe

On Parameterized Independent Feedback Vertex Set [chapter]

Neeldhara Misra, Geevarghese Philip, Venkatesh Raman, Saket Saurabh
2011 Lecture Notes in Computer Science  
We investigate a generalization of the classical Feedback Vertex Set (FVS) problem from the point of view of parameterized algorithms.  ...  We show that IFVS can be solved in time O(5 k n O(1) ) time where n is the number of vertices in the input graph G, and obtain a cubic (O(k 3 )) kernel for the problem.  ...  A Cubic Kernel In this section we describe a kernelization algorithm which yields a kernel of size O(k 3 ) for the IFVS problem.  ... 
doi:10.1007/978-3-642-22685-4_9 fatcat:enmtafu2unf6hb2kgesdvr2pum

Page 5811 of Mathematical Reviews Vol. , Issue 96j [page]

1996 Mathematical Reviews  
new bound on the feedback vertex sets in cubic graphs.  ...  Summary: “For a graph G, a subset of vertices D is a k-dominating set of G if for each vertex x not in D, x is at distance at most k from some vertex in D.  ... 

An Improved FPT Algorithm and Quadratic Kernel for Pathwidth One Vertex Deletion [chapter]

Marek Cygan, Marcin Pilipczuk, Michał Pilipczuk, Jakub Onufry Wojtaszczyk
2010 Lecture Notes in Computer Science  
In this paper we improve these results by showing a quadratic kernel and a 4.65 k n O(1) algorithm.  ...  Recently Philip et al. [14] initiated the study of the parameterized complexity of POVD and have shown a quartic kernel and a 7 k n O(1) algorithm.  ...  One of the most extensively studied problems in the parameterized complexity community, FEEDBACK VERTEX SET, asks for a set of vertices of size at most k such that their deletion results in a graph of  ... 
doi:10.1007/978-3-642-17493-3_11 fatcat:orqu3ziug5ez7geggcmlmvfhua

Parameterized Aspects of Triangle Enumeration [article]

Matthias Bentert, Till Fluschnik, André Nichterlein, Rolf Niedermeier
2018 arXiv   pre-print
tuned for large graphs).  ...  Motivated by the fact that the worst-case running time is cubic, we perform a systematic parameterized complexity study of triangle enumeration.  ...  We thank Mark Ortmann for fruitful discussions on obstacles for practical algorithms.  ... 
arXiv:1702.06548v4 fatcat:b7z7jh6tjngxfazvfppoqkiu4y

Vertex Cover Kernelization Revisited

Bart M. P. Jansen, Hans L. Bodlaender
2012 Theory of Computing Systems  
We give a kernel for Vertex Cover with a number of vertices that is cubic in fvs(G): an instance (G,X,k) of Vertex Cover, where X is a feedback vertex set for G, can be transformed in polynomial time into  ...  In this paper we consider the Vertex Cover problem with a different parameter, the size fvs(G) of a minimum feedback vertex set for G.  ...  Acknowledgements We are grateful to the anonymous referees, whose suggestions significantly improved the exposition of our results and decreased the running time of the kernelization procedure.  ... 
doi:10.1007/s00224-012-9393-4 fatcat:qfkejrlmfrfq5kwihwau2ieqwi

On the Parameterized Complexity of Reconfiguration Problems

Amer E. Mouawad, Naomi Nishimura, Venkatesh Raman, Narges Simjour, Akira Suzuki
2016 Algorithmica  
Hitting Set and Feedback Vertex Set, all parameterized by k.  ...  For most of the results in this paper, S and T are subsets of vertices of a given graph and a reconfiguration step adds or deletes a vertex.  ...  The second author wishes to thank Marcin Kamiński for suggesting the examination of reconfiguration in the parameterized setting.  ... 
doi:10.1007/s00453-016-0159-2 fatcat:qretakdj4nhhtgipvk4fyjlhsq

On the Parameterized Complexity of Reconfiguration Problems [chapter]

Amer E. Mouawad, Naomi Nishimura, Venkatesh Raman, Narges Simjour, Akira Suzuki
2013 Lecture Notes in Computer Science  
Hitting Set and Feedback Vertex Set, all parameterized by k.  ...  For most of the results in this paper, S and T are subsets of vertices of a given graph and a reconfiguration step adds or deletes a vertex.  ...  The second author wishes to thank Marcin Kamiński for suggesting the examination of reconfiguration in the parameterized setting.  ... 
doi:10.1007/978-3-319-03898-8_24 fatcat:fg5zfk6ov5egtda4i2hqvhbocu

On Parameterized Independent Feedback Vertex Set

Neeldhara Misra, Geevarghese Philip, Venkatesh Raman, Saket Saurabh
2012 Theoretical Computer Science  
We investigate a generalization of the classical Feedback Vertex Set (FVS) problem from the point of view of parameterized algorithms.  ...  We show that IFVS can be solved in O(5 k n O(1) ) time, where n is the number of vertices in the input graph G, and obtain a cubic (O(k 3 )) kernel for the problem.  ...  A cubic kernel In this section, we describe a kernelization algorithm which yields a kernel of size O(k 3 ) for the IFVS problem.  ... 
doi:10.1016/j.tcs.2012.02.012 fatcat:rgtgkd2snnbf5mb3q2rymiraky

Parameterized Aspects of Triangle Enumeration [chapter]

Matthias Bentert, Till Fluschnik, André Nichterlein, Rolf Niedermeier
2017 Lecture Notes in Computer Science  
In particular, we provide enumerative problem kernels with respect to the parameters "feedback edge number" and "vertex deletion distance to d-degeneracy".  ...  Partially based on the corresponding data reduction algorithms, we provide "faster" algorithms for parameters such as feedback edge number, vertex deletion distance to d-degeneracy (also with additional  ...  feedback vertex set (distance to 1-degenerate graphs).  ... 
doi:10.1007/978-3-662-55751-8_9 fatcat:yuuptmzjvjearabsr7kzg6vdui
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