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

Stefan Kratsch
2016 arXiv   pre-print
We complement the work of Garg and Philip by proving that 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).  ... 
arXiv:1611.06795v1 fatcat:e34ito3nize7xglglwjgkmrubq

Smaller parameters for vertex cover kernelization [article]

Eva-Maria C. Hols, Stefan Kratsch
2017 arXiv   pre-print
We revisit the topic of 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] .  ... 
arXiv:1711.04604v1 fatcat:ytmrvrazhrgiti3wny2ek2jvyq

Towards fully multivariate algorithmics: Parameter ecology and the deconstruction of computational complexity

Michael R. Fellows, Bart M.P. Jansen, Frances Rosamond
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?  ... 
doi:10.1016/j.ejc.2012.04.008 fatcat:uzjyjuxec5hvlcep3jsf3mcupu

Kernelization – Preprocessing with a Guarantee [chapter]

Daniel Lokshtanov, Neeldhara Misra, Saket Saurabh
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.  ... 
doi:10.1007/978-3-642-30891-8_10 fatcat:ihw5qxdnffdybk4f2c4lhds3pe

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

Syed Mohammad Meesum, Fahad Panolan, Saket Saurabh, Meirav Zehavi
2017 arXiv   pre-print
The question of the existence of 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  ... 
arXiv:1705.02822v2 fatcat:ajwp4pspqfchhpucwykzifefhm

Parameterized Algorithmics and Computational Experiments for Finding 2-Clubs [chapter]

Sepp Hartung, Christian Komusiewicz, André Nichterlein
2012 Lecture Notes in Computer Science  
On the negative side, we first show that unless NP ⊆ coNP/poly, 2-Club does not admit 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.  ... 
doi:10.1007/978-3-642-33293-7_22 fatcat:5hbgpg6wuzbcbetfwpivstuxle

Parameterized Algorithmics and Computational Experiments for Finding 2-Clubs

Sepp Hartung, Christian Komusiewicz, André Nichterlein
2015 Journal of Graph Algorithms and Applications  
On the negative side, we first show that unless NP ⊆ coNP/poly, 2-Club does not admit 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.  ... 
doi:10.7155/jgaa.00352 fatcat:5xnvb6pycbf2rjuz35ju7p7tgq

Representative sets and irrelevant vertices: New tools for kernelization [article]

Stefan Kratsch, Magnus Wahlström
2012 arXiv   pre-print
The existence of 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] .  ... 
arXiv:1111.2195v2 fatcat:ienvm4zagndaxhew246stssnka

Dynamic Parameterized Problems

R. Krithika, Abhishek Sahu, Prafullkumar Tale
2017 Algorithmica  
Then, we show that Dynamic Feedback 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.  ... 
doi:10.1007/s00453-017-0349-6 fatcat:w42snepu6zc3helts3t4xyum6a

The Curse of Connectivity: t-Total Vertex (Edge) Cover [chapter]

Henning Fernau, Fedor V. Fomin, Geevarghese Philip, Saket Saurabh
2010 Lecture Notes in Computer Science  
has no 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.  ... 
doi:10.1007/978-3-642-14031-0_6 fatcat:njhacuvt45gnzarjvficrqbkxy

Abusing the Tutte Matrix: An Algebraic Instance Compression for the K-set-cycle Problem [article]

Magnus Wahlström
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.  ... 
arXiv:1301.1517v1 fatcat:k4hpsk5n3za33as54tfg7c25ey

Parameterized algorithms of fundamental NP-hard problems: a survey

Wenjun Li, Yang Ding, Yongjie Yang, R. Simon Sherratt, Jong Hyuk Park, Jin Wang
2020 Human-Centric Computing and Information Sciences  
, Feedback 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.  ... 
doi:10.1186/s13673-020-00226-w fatcat:2ncywbdlnrdtzliiz4tyi76df4

A Parameterized Complexity Tutorial [chapter]

Rod Downey
2012 Lecture Notes in Computer Science  
The article was prepared 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.  ... 
doi:10.1007/978-3-642-28332-1_4 fatcat:plrubtmiebh3pkhzzov7k3rmby

Lower bounds on kernelization

Neeldhara Misra, Venkatesh Raman, Saket Saurabh
2011 Discrete Optimization  
So 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.  ... 
doi:10.1016/j.disopt.2010.10.001 fatcat:kvbpwrhvjrf53fm5bisef7tg64

Efficient Implementation of the BSP/CGM Parallel Vertex Cover FPT Algorithm [chapter]

Erik J. Hanashiro, Henrique Mongelli, Siang W. Song
2004 Lecture Notes in Computer Science  
One promising method to treat some intractable problems is by considering the so-called 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.  ... 
doi:10.1007/978-3-540-24838-5_19 fatcat:xlzbazha6vbezk6qszxhdafpni
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