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A Cubic Kernel for Feedback Vertex Set and Loop Cutset

2009
*
Theory of Computing Systems
*

This

doi:10.1007/s00224-009-9234-2
fatcat:cwq2aws44bhmhd4r3qxbdqcvoq
*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. ...##
###
A Cubic Kernel for Feedback Vertex Set
[chapter]

*
STACS 2007
*

This

doi:10.1007/978-3-540-70918-3_28
dblp:conf/stacs/Bodlaender07
fatcat:blfcrzsppnfnrdn5fb2zo2cbbm
*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 . ...##
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Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter

2011
*
Symposium on Theoretical Aspects of Computer Science
*

be reduced to

doi:10.4230/lipics.stacs.2011.177
dblp:conf/stacs/JansenB11
fatcat:v6mfndllc5bsxctpwhranx52wa
*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? ...##
###
A quadratic kernel for feedback vertex set
[chapter]

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. ...

##
###
A 4k2kernel for feedback vertex set

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 . ...

##
###
Safe Approximation and Its Relation to Kernelization
[chapter]

2012
*
Lecture Notes in Computer Science
*

We introduce

doi:10.1007/978-3-642-28050-4_14
fatcat:2c7xqcqavrh7jae74ms2ndbowe
*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. ...##
###
On Parameterized Independent Feedback Vertex Set
[chapter]

2011
*
Lecture Notes in Computer Science
*

We investigate

doi:10.1007/978-3-642-22685-4_9
fatcat:enmtafu2unf6hb2kgesdvr2pum
*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. ...##
###
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]

2010
*
Lecture Notes in Computer Science
*

In this paper we improve these results by showing

doi:10.1007/978-3-642-17493-3_11
fatcat:orqu3ziug5ez7geggcmlmvfhua
*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 ...##
###
Parameterized Aspects of Triangle Enumeration
[article]

2018
*
arXiv
*
pre-print

tuned

arXiv:1702.06548v4
fatcat:b7z7jh6tjngxfazvfppoqkiu4y
*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. ...##
###
Vertex Cover Kernelization Revisited

2012
*
Theory of Computing Systems
*

We give

doi:10.1007/s00224-012-9393-4
fatcat:qfkejrlmfrfq5kwihwau2ieqwi
*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. ...##
###
On the Parameterized Complexity of Reconfiguration Problems

2016
*
Algorithmica
*

Hitting

doi:10.1007/s00453-016-0159-2
fatcat:qretakdj4nhhtgipvk4fyjlhsq
*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*. ...##
###
On the Parameterized Complexity of Reconfiguration Problems
[chapter]

2013
*
Lecture Notes in Computer Science
*

Hitting

doi:10.1007/978-3-319-03898-8_24
fatcat:fg5zfk6ov5egtda4i2hqvhbocu
*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*. ...##
###
On Parameterized Independent Feedback Vertex Set

2012
*
Theoretical Computer Science
*

We investigate

doi:10.1016/j.tcs.2012.02.012
fatcat:rgtgkd2snnbf5mb3q2rymiraky
*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. ...##
###
Parameterized Aspects of Triangle Enumeration
[chapter]

2017
*
Lecture Notes in Computer Science
*

In particular, we provide enumerative problem

doi:10.1007/978-3-662-55751-8_9
fatcat:yuuptmzjvjearabsr7kzg6vdui
*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). ...
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