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Polynomial Kernels for Hard Problems on Disk Graphs [chapter]

Bart Jansen
2010 Lecture Notes in Computer Science  
Finally we prove that H-Matching 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.  ... 
doi:10.1007/978-3-642-13731-0_30 fatcat:twjv4g5nkzd7lkcfgcncx2i5ee

Parameterized Complexity of Geometric Problems

P. Giannopoulos, C. Knauer, S. Whitesides
2007 Computer journal  
This paper surveys parameterized complexity results 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.  ... 
doi:10.1093/comjnl/bxm053 fatcat:ohzdxo2ehbcgrawsgmzdcvz5xq

Parameterized Complexity of Graph Partitioning into Connected Clusters [article]

Ankit Abhinav, Susobhan Bandopadhyay, Aritra Banik, Saket Saurabh
2022 arXiv   pre-print
Here we study both the 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.  ... 
arXiv:2202.12042v1 fatcat:xujq54huqfgupfuvnrrni2asgi

A Survey on the k-Path Vertex Cover Problem [article]

Jianhua Tu
2022 arXiv   pre-print
Given a 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).  ... 
arXiv:2201.03397v2 fatcat:v5caz3jmcjcxzhia2476odhy4a

A Survey on the k-Path Vertex Cover Problem

Jianhua Tu
2022 Axioms  
Given an integer k ≥ 2, a k-path is a path 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.  ... 
doi:10.3390/axioms11050191 fatcat:ympjlwzforexnn7khqpee3wlaq

Geometric separation and exact solutions for the parameterized independent set problem on disk graphs

Jochen Alber, Jiří Fiala
2004 Journal of Algorithms  
The results are based 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  ... 
doi:10.1016/j.jalgor.2003.10.001 fatcat:3meqsh4birhtdesa4pbygqw6we

Alternative parameterizations of Metric Dimension [article]

Gregory Gutin, M. S. Ramanujan, Felix Reidl, Magnus Wahlström
2018 arXiv   pre-print
Our reduction also gives evidence 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.  ... 
arXiv:1804.10670v1 fatcat:j7fxq56umfgczhw7baeisrqir4

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]

Jochen Alber, Jiří Fiala
2002 Foundations of Information Technology in the Era of Network and Mobile Computing  
The results are based 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.  ... 
doi:10.1007/978-0-387-35608-2_3 dblp:conf/ifipTCS/AlberF02 fatcat:uwl6jfili5avvhw5x6wfswrimy

What's Next? Future Directions in Parameterized Complexity [chapter]

Dániel Marx
2012 Lecture Notes in Computer Science  
In particular, we set up a framework 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.  ... 
doi:10.1007/978-3-642-30891-8_20 fatcat:enftdcd27rd5jmglgw4khqffty

1-Extendability of independent sets [article]

Pierre Bergé, Anthony Busson, Carl Feghali, Rémi Watrigant
2022 arXiv   pre-print
subcubic planar 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.  ... 
arXiv:2204.05809v1 fatcat:zxhfk5j2erbotcrnjg3b52n4hq

Computing Maximum Matchings in Temporal Graphs

George B. Mertzios, Hendrik Molter, Rolf Niedermeier, Viktor Zamaraev, Philipp Zschoche, Markus Bläser, Christophe Paul
2020 Symposium on Theoretical Aspects of Computer Science  
To cope with this computational 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.  ... 
doi:10.4230/lipics.stacs.2020.27 dblp:conf/stacs/MertziosMNZZ20 fatcat:vix6zw2cfnhrdnh5n5ligun2lu

The Computer Journal Special Issue on Parameterized Complexity: Foreword by the Guest Editors

R. G. Downey, M. R. Fellows, M. A. Langston
2007 Computer journal  
Local search heuristics 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?  ... 
doi:10.1093/comjnl/bxm111 fatcat:jhqptpkwj5c63a6t6xuqgj3dju

(Meta) Kernelization

Hans L. Bodlaender, Fedor V. Fomin, Daniel Lokshtanov, Eelko Penninkx, Saket Saurabh, Dimitrios M. Thilikos
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] .  ... 
doi:10.1145/2973749 fatcat:uj2dw4sl4vdebbrmrry6f27rvi

(Meta) Kernelization

Hans L. Bodlaender, Fedor V. Fomin, Daniel Lokshtanov, Eelko Penninkx, Saket Saurabh, Dimitrios M. Thilikos
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] .  ... 
doi:10.1109/focs.2009.46 dblp:conf/focs/BodlaenderFLPST09 fatcat:563wr5dibvf6nooyp4ntyxcwiq
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