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A randomized polynomial kernelization for Vertex Cover with a smaller parameter
[article]

2016
*
arXiv
*
pre-print

We complement the work of Garg and Philip by proving that

arXiv:1611.06795v1
fatcat:e34ito3nize7xglglwjgkmrubq
*Vertex**Cover*admits*a**randomized**polynomial**kernelization*in terms of ℓ, i.e., an efficient preprocessing to size*polynomial*in ℓ. ... They prove that 2LP(G)-MM(G) is*a*lower bound*for*the*vertex**cover*size of G, where MM(G) is the size of*a*largest matching of G, and proceed to study*parameter*ℓ=k-(2LP(G)-MM(G)). ... Conclusion We have presented*a**randomized**polynomial**kernelization**for**vertex**cover*(k − (2lp − mm)) by giving*a*(*randomized*)*polynomial**parameter*transformation to*vertex**cover*(k − mm). ...##
###
Smaller parameters for vertex cover kernelization
[article]

2017
*
arXiv
*
pre-print

We revisit the topic of

arXiv:1711.04604v1
fatcat:ytmrvrazhrgiti3wny2ek2jvyq
*polynomial**kernels**for**Vertex**Cover*relative to structural*parameters*. ... Combined*with*existing*randomized**polynomial**kernelizations*this leads to*randomized**polynomial**kernelizations**for*modulators to d-quasi-bipartite and d-quasi-integral graphs. ... by using*a**randomized**polynomial**kernelization**for**vertex**cover**parameterized*by an odd cycle transversal [15] . ...##
###
Towards fully multivariate algorithmics: Parameter ecology and the deconstruction of computational complexity

2013
*
European journal of combinatorics (Print)
*

*polynomial*

*kernels*. ... An extensive overview of recent advances on this front is presented

*for*the

*Vertex*

*Cover*problem. ... Is there

*a*structural

*parameter*

*smaller*than feedback

*vertex*number

*with*respect to which Feedback

*Vertex*Set admits

*a*

*polynomial*

*kernel*? ...

##
###
Kernelization – Preprocessing with a Guarantee
[chapter]

2012
*
Lecture Notes in Computer Science
*

*A*

*kernelization*algorithm is

*a*preprocessing algorithm which simplifies the instances given as input in

*polynomial*time, and the extent of simplification desired is quantified

*with*the help of the additional ... The framework of

*parameterized*complexity turns out to be particularly suitable

*for*

*a*mathematical analysis of pre-processing heuristics. ... In the

*parameterized*

*vertex*

*cover*problem, we call it p-

*Vertex*

*Cover*the

*parameter*is an integer k and we ask whether the input graph has

*a*

*vertex*

*cover*of size at most k. ...

##
###
Rank Vertex Cover as a Natural Problem for Algebraic Compression
[article]

2017
*
arXiv
*
pre-print

The question of the existence of

arXiv:1705.02822v2
fatcat:ajwp4pspqfchhpucwykzifefhm
*a**polynomial**kernelization*of the*Vertex**Cover*Above LP problem has been*a*longstanding, notorious open problem in*Parameterized*Complexity. ... Here, the input consists of*a*graph G,*a**parameter*k, and*a*bijection between V (G) and the set of columns of*a*representation of*a*matriod M, and the objective is to find*a**vertex**cover*whose rank is ... More precisely, we develop*a**polynomial*-time*randomized*algorithm that given an instance of*Vertex**Cover*Above LP*with**parameter*k and ε > 0,*with*probability at least 1 − ε outputs an equivalent instance ...##
###
Parameterized Algorithmics and Computational Experiments for Finding 2-Clubs
[chapter]

2012
*
Lecture Notes in Computer Science
*

On the negative side, we first show that unless NP ⊆ coNP/poly, 2-Club does not admit

doi:10.1007/978-3-642-33293-7_22
fatcat:5hbgpg6wuzbcbetfwpivstuxle
*a**polynomial*-size problem*kernel**with*respect to the size of*a**vertex**cover*of G. ... On the positive side, we give*polynomial*-size problem*kernels**for*the*parameters*feedback edge set size of G and size of*a*cluster editing set of G and present*a*direct combinatorial algorithm*for*the ... Proof: We give*a**polynomial*-time and -*parameter*reduction [9] from Clique*parameterized*by*vertex**cover*to 2-Club*parameterized*by*vertex**cover*. ...##
###
Parameterized Algorithmics and Computational Experiments for Finding 2-Clubs

2015
*
Journal of Graph Algorithms and Applications
*

On the negative side, we first show that unless NP ⊆ coNP/poly, 2-Club does not admit

doi:10.7155/jgaa.00352
fatcat:5xnvb6pycbf2rjuz35ju7p7tgq
*a**polynomial*-size problem*kernel**with*respect to the size of*a**vertex**cover*of G. ... On the positive side, we give*polynomial*-size problem*kernels**for*the*parameters*feedback edge set size of G and size of*a*cluster editing set of G and present*a*direct combinatorial algorithm*for*the ... Proof: We give*a**polynomial*-time and -*parameter*reduction [9] from Clique*parameterized*by*vertex**cover*to 2-Club*parameterized*by*vertex**cover*. ...##
###
Representative sets and irrelevant vertices: New tools for kernelization
[article]

2012
*
arXiv
*
pre-print

The existence of

arXiv:1111.2195v2
fatcat:ienvm4zagndaxhew246stssnka
*a**polynomial**kernel**for*Odd Cycle Transversal was*a*notorious open problem in*parameterized*complexity. ... Recently, this was settled by the present authors (Kratsch and Wahlström, SODA 2012),*with**a**randomized**polynomial**kernel**for*the problem, using matroid theory to encode flow questions over*a*set of terminals ... The authors are grateful to Dániel Marx*for*comments on an early draft of the paper, and to Saket Saurabh*for*an early version of [32] . ...##
###
Dynamic Parameterized Problems

2017
*
Algorithmica
*

Then, we show that Dynamic Feedback

doi:10.1007/s00453-017-0349-6
fatcat:w42snepu6zc3helts3t4xyum6a
*Vertex*Set admits*a**randomized*algorithm*with*1.6667 k n O(1) running time. ... Dynamic*Vertex**Cover*admits*a**kernel**with*at most 2k vertices and at most k edges. Proof. Consider an instance (G, G , S, k, r) of Dynamic*Vertex**Cover*. ... We are also thankful to the reviewers*for*their suggestions that strengthened the results*for*Dynamic Π-Deletion. ...##
###
The Curse of Connectivity: t-Total Vertex (Edge) Cover
[chapter]

2010
*
Lecture Notes in Computer Science
*

has no

doi:10.1007/978-3-642-14031-0_6
fatcat:njhacuvt45gnzarjvficrqbkxy
*polynomial**kernel*unless the*Polynomial*Hierarchy collapses to the third level; -*for*every t ≥ 2, t-TOTAL EDGE*COVER*has*a*linear*vertex**kernel*of size t+1 t k. ... Our no-poly-*kernel*result*for*t-TOTAL*VERTEX**COVER*, and the known NP-hardness result*for*t-TOTAL EDGE*COVER*, are in stark contrast to the fact that k-*VERTEX**COVER*has*a*2k*vertex**kernel*, and that k-EDGE ... We complement the no-*polynomial*-*kernel*result*with*results on the fixed-*parameter*tractability of t-TOTAL*VERTEX**COVER*and t-TOTAL EDGE*COVER*when*parameterized*by the solution size. ...##
###
Abusing the Tutte Matrix: An Algebraic Instance Compression for the K-set-cycle Problem
[article]

2013
*
arXiv
*
pre-print

*kernelization*(as

*a*

*polynomial*

*kernelization*

*for*

*a*problem in NP necessarily implies

*a*small witness). ... Furthermore, our approach is open

*for*treatment by classical algebraic tools (e.g., Gaussian elimination), and we show that it leads to

*a*

*polynomial*compression of the problem, i.e.,

*a*

*polynomial*-time ... Acknowledgements The author is grateful to Thore Husfeldt and Stefan Kratsch

*for*rewarding discussions, and to an anonymous reviewer

*for*suggesting improvements to the paper. ...

##
###
Parameterized algorithms of fundamental NP-hard problems: a survey

2020
*
Human-Centric Computing and Information Sciences
*

, Feedback

doi:10.1186/s13673-020-00226-w
fatcat:2ncywbdlnrdtzliiz4tyi76df4
*Vertex*Set, Hyperplane*Cover*,*Vertex**Cover*, Packing and Matching problems. ... We give an overview on*parameterized*algorithms*for*some fundamental NP-hard problems, including MaxSAT, Maximum Internal Spanning Trees, Maximum Internal Out-Branching, Planar (Connected) Dominating Set ... Acknowledgements The authors would like to thank the reviewers*for*their valuable comments which improve the quality of this paper dramatically. ...##
###
A Parameterized Complexity Tutorial
[chapter]

2012
*
Lecture Notes in Computer Science
*

The article was prepared

doi:10.1007/978-3-642-28332-1_4
fatcat:plrubtmiebh3pkhzzov7k3rmby
*for*the LATA 2012 conference where I will be presenting two one and half hour lecctures*for**a*short tutorial on*parameterized*complexity. ... We consider*a**smaller**vertex**cover*C as*a*modification of the larger*vertex**cover*C. ... The compression routine takes*a*graph G and*a**vertex**cover*C*for*G and returns*a**smaller**vertex**cover**for*G if there is one, otherwise, it returns C unchanged. ...##
###
Lower bounds on kernelization

2011
*
Discrete Optimization
*

So

doi:10.1016/j.disopt.2010.10.001
fatcat:kvbpwrhvjrf53fm5bisef7tg64
*a*natural question*for*any concrete FPT problem is whether it admits*polynomial*time*kernelization*to*a*problem*kernel*that in the worst case is bounded by*a**polynomial*function of the*parameter*. ...*Parameterized*complexity provides*a*natural framework to analyse preprocessing algorithms. In*a**parameterized*problem, every instance x comes*with**a*positive integer, or*parameter*, k. ... Consider,*for*instance, graphs that have "small"*vertex**covers*. These are graphs that have*vertex**cover*of size t (or*smaller*)*for*an arbitrary but fixed t. ...##
###
Efficient Implementation of the BSP/CGM Parallel Vertex Cover FPT Algorithm
[chapter]

2004
*
Lecture Notes in Computer Science
*

One promising method to treat some intractable problems is by considering the so-called

doi:10.1007/978-3-540-24838-5_19
fatcat:xlzbazha6vbezk6qszxhdafpni
*Parameterized*Complexity that divides the problem input into*a*main part and*a**parameter*. ... Furthermore,*for*three graphs, we obtained*smaller*sizes*for*the*cover*. ⋆ Partially supported by FAPESP ... Cáceres (UFMS)*for*his assistance, the Institute of Computing/UNICAMP*for*giving the permission to use the machines, and finally the referees*for*their helpful comments. ...
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