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Polynomial Kernels for Hard Problems on Disk Graphs
[chapter]

2010
*
Lecture Notes in Computer Science
*

Finally we prove that H-Matching

doi:10.1007/978-3-642-13731-0_30
fatcat:twjv4g5nkzd7lkcfgcncx2i5ee
*on*unit-*disk**graphs*has a linear-vertex*kernel**for*every fixed*graph*H. ... Recent breakthroughs show that many*graph**problems*admit small*polynomial**kernels*when restricted to sparse*graph*classes such as planar*graphs*, bounded-genus*graphs*or H-minor-free*graphs*. ... Acknowledgements I would like to thank Manu Basavaraju, Hans Bodlaender, Marc van Kreveld, Erik Jan van Leeuwen and Jan van Leeuwen*for*insightful discussions. ...##
###
Parameterized Complexity of Geometric Problems

2007
*
Computer journal
*

This paper surveys parameterized complexity results

doi:10.1093/comjnl/bxm053
fatcat:ohzdxo2ehbcgrawsgmzdcvz5xq
*for**hard*geometric algorithmic*problems*. ... It includes fixed-parameter tractable*problems*in*graph*drawing, geometric*graphs*, geometric covering and several other areas, together with an overview of the algorithmic techniques used. ... ACKNOWLEDGEMENTS We would like to thank Mike Fellows*for*his support, comments and ideas in the course of writing this survey. ...##
###
Parameterized Complexity of Graph Partitioning into Connected Clusters
[article]

2022
*
arXiv
*
pre-print

Here we study both the

arXiv:2202.12042v1
fatcat:xujq54huqfgupfuvnrrni2asgi
*problems**for*q=2 and prove that BCP_q*for*q≥ 2 is W[1]-*hard*. We also show that BCP_2 is unlikely to have a*polynomial**kernel**on*the class of planar*graphs*. ... We design another FPT algorithm and a*polynomial**kernel**on*the class of unit*disk**graphs*parameterized by min(n_1,n_2). ... No*Polynomial**Kernels**for*BCP 2 in Planar*Graph*In this section, we prove that no*polynomial*sized*kernel*exists*for*BCP 2*on*planar*graphs*. ...##
###
A Survey on the k-Path Vertex Cover Problem
[article]

2022
*
arXiv
*
pre-print

Given a

arXiv:2201.03397v2
fatcat:v5caz3jmcjcxzhia2476odhy4a
*graph*G=(V,E) and a positive integer k≥2, a k-path vertex cover is a subset of vertices F such that every path*on*k vertices in G contains at least*one*vertex from F. ... In the k-path vertex cover*problem*, it is required to find a minimum k-path vertex cover in a given*graph*. ... Acknowledgments The work was supported by Research Foundation*for*Advanced Talents of Beijing Technology and Business University (No. 19008021187). ...##
###
A Survey on the k-Path Vertex Cover Problem

2022
*
Axioms
*

Given an integer k ≥ 2, a k-path is a path

doi:10.3390/axioms11050191
fatcat:ympjlwzforexnn7khqpee3wlaq
*on*k vertices. A set of vertices in a*graph*G is called a k-path vertex cover if it includes at least*one*vertex of every k-path of G. ... In the k-path vertex cover*problem*, the goal is to find a minimum k-path vertex cover in a given*graph*. ... [47] proved that WC-MinVCP 3 is NP-*hard*even in unit*disk**graphs*. ...##
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Geometric separation and exact solutions for the parameterized independent set problem on disk graphs

2004
*
Journal of Algorithms
*

The results are based

doi:10.1016/j.jalgor.2003.10.001
fatcat:3meqsh4birhtdesa4pbygqw6we
*on*a new "geometric 0-separator theorem" which holds*for*all*disk**graphs*of bounded radius. ... The presented algorithm then performs, in a first step, a "geometric*problem**kernelization*" and, in a second step, uses divide-and-conquer based*on*our geometric separator theorem. ... We thank Thomas Erlebach and Rolf Niedermeier*for*introducing us to the*problem*and*for*fruitful discussion and valuable comments ...##
###
Alternative parameterizations of Metric Dimension
[article]

2018
*
arXiv
*
pre-print

Our reduction also gives evidence

arXiv:1804.10670v1
fatcat:j7fxq56umfgczhw7baeisrqir4
*for*non-existence of*polynomial*Turing*kernels*. ... The Metric Dimension*problem*, i.e. deciding whether md(G)< k, is NP-complete even*for*interval*graphs*(Foucaud et al., 2017). ... Hoffmann and Wanke [19] extended the tractability results to a subclass of unit*disk**graphs*, while Foucaud et al. [12] showed that this*problem*is NP-complete*on*interval*graphs*. ...##
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Page 4863 of Mathematical Reviews Vol. , Issue 2004f
[page]

2004
*
Mathematical Reviews
*

Determining whether a

*graph*is a unit*disk**graph*is NP-*hard*and not known to be in NP. There is a poly-time solution to the*problem*of finding the maximum clique in a unit*disk**graph*. ... . (2) A unit*disk**graph*is a*graph*where the vertices are*graphs*of unit radius, and two of them have an edge if they intersect. ...##
###
Geometric Separation and Exact Solutions for the Parameterized Independent Set Problem on Disk Graphs
[chapter]

2002
*
Foundations of Information Technology in the Era of Network and Mobile Computing
*

The results are based

doi:10.1007/978-0-387-35608-2_3
dblp:conf/ifipTCS/AlberF02
fatcat:uwl6jfili5avvhw5x6wfswrimy
*on*a new "geometric √ ·-separator theorem" which holds*for*all*disk**graphs*of bounded radius ratio. ... Our techniques can be extended to various other*graph**problems*, such as dominating set, to obtain similar results*for**disk**graphs*of bounded radius ratio. ... We thank Thomas Erlebach and Rolf Niedermeier*for*introducing us to the*problem*and*for*fruitful discussion and valuable comments during this project. ...##
###
What's Next? Future Directions in Parameterized Complexity
[chapter]

2012
*
Lecture Notes in Computer Science
*

In particular, we set up a framework

doi:10.1007/978-3-642-30891-8_20
fatcat:enftdcd27rd5jmglgw4khqffty
*for*understanding which*problems*can be solved by branching algorithms. ... Finally, we discuss that the domain of directed*graph**problems*is a challenging area which can potentially see significant progress in the following years. ...*Hard**problems**on*undirected*graphs*are often studied*on*particular subclasses of*graphs*:*on*planar*graphs*, interval*graphs*, bounded-treewidth*graphs*, etc. ...##
###
1-Extendability of independent sets
[article]

2022
*
arXiv
*
pre-print

subcubic planar

arXiv:2204.05809v1
fatcat:zxhfk5j2erbotcrnjg3b52n4hq
*graphs*and in unit*disk**graphs*(which is a natural model*for*wireless networks). ... Maximum Independent Set), we show that, interestingly, there exist 1-extendable*graphs**for*which Maximum Independent Set is NP-*hard*. Finally, we investigate a parameterized version of 1-extendability. ... Theorem 5. 1-Extendability is NP-*hard*, even*on*unit*disk**graphs*. ...##
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Computing Maximum Matchings in Temporal Graphs

2020
*
Symposium on Theoretical Aspects of Computer Science
*

To cope with this computational

doi:10.4230/lipics.stacs.2020.27
dblp:conf/stacs/MertziosMNZZ20
fatcat:vix6zw2cfnhrdnh5n5ligun2lu
*hardness*, we mainly focus*on*fixed-parameter algorithms with respect to natural parameters, as well as*on**polynomial*-time approximation algorithms. ... We prove strong computational*hardness*results*for*Maximum Temporal Matching, even*for*elementary cases. ... [9]*for*unit*disk**graphs*. ...##
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The Computer Journal Special Issue on Parameterized Complexity: Foreword by the Guest Editors

2007
*
Computer journal
*

Local search heuristics

doi:10.1093/comjnl/bxm111
fatcat:jhqptpkwj5c63a6t6xuqgj3dju
*for**hard**problems*utilize a*polynomial*- sized neighborhood N (S) of a ... But the*problem*itself is both NP-*hard*and W-*hard*. But what is that structure, that allows the*problem*to be solved exactly in practice*for*these huge*graphs*? ...##
###
(Meta) Kernelization

2016
*
Journal of the ACM
*

*Polynomial*time preprocessing to reduce instance size is

*one*of the most commonly deployed heuristics to tackle computationally

*hard*

*problems*. ... Our theorems unify and extend all previously known

*kernelization*results

*for*planar

*graph*

*problems*. ... We thank Jiong Guo

*for*sending us the complete version of [38] . ...

##
###
(Meta) Kernelization

2009
*
2009 50th Annual IEEE Symposium on Foundations of Computer Science
*

*Polynomial*time preprocessing to reduce instance size is

*one*of the most commonly deployed heuristics to tackle computationally

*hard*

*problems*. ... Our theorems unify and extend all previously known

*kernelization*results

*for*planar

*graph*

*problems*. ... We thank Jiong Guo

*for*sending us the complete version of [38] . ...

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