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Bidimensionality and Kernels [chapter]

Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, Dimitrios M. Thilikos
2010 Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms  
We show that "essentially" all bidimensional problems not only have sub-exponential time algorithms and PTASs but they also have linear kernels, affirmatively answering an open question from [J.  ...  Demaine and Hajiaghayi [SODA 2005 ] extended the theory to obtain polynomial time approximation schemes (PTASs) for bidimensional problems.  ...  In particular, does DOMINATING SET and CONNECTED DOMINATING SET?  ... 
doi:10.1137/1.9781611973075.43 dblp:conf/soda/FominLST10 fatcat:stys2rqdjzapvp5viwv3ojvfoy

Bidimensionality and Kernels [article]

Fedor V. Fomin and Daniel Lokshtanov and Saket Saurabh and Dimitrios M. Thilikos
2020 arXiv   pre-print
Demaine and M.Hajiaghayi, Bidimensionality: new connections between FPT algorithms and PTASs, in Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, 2005, pp.590--601  ...  Bidimensionality Theory was introduced by [E.D. Demaine, F.V. Fomin, M.Hajiaghayi, and D.M. Thilikos. Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs, J.  ...  We define S 1 = S ∩ V (G 1 ) and S = S \ S 1 and we know that (G 1 ⊕ F, S 1 ∪ S ) |= ε. Notice that |S 1 | + |S | = |S 1 ∪ S | ≤ k.  ... 
arXiv:1606.05689v3 fatcat:472cpmaxlfailcqrtajs32sl5i

Bidimensionality and Geometric Graphs [chapter]

Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh
2012 Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms  
In this paper we use several of the key ideas from Bidimensionality to give a new generic approach to design EPTASs and subexponential time parameterized algorithms for problems on classes of graphs which  ...  Bidimensionality and Separability.  ...  This means that the two problems neither admit EPTASs nor FPT algorithms unless FPT=W [1] , a complexity collapse considered very unlikely.  ... 
doi:10.1137/1.9781611973099.124 dblp:conf/soda/FominLS12 fatcat:s7q42vkczjbp7drdztgo77ool4

Bidimensionality and Geometric Graphs [article]

Fedor V. Fomin and Daniel Lokshtanov and Saket Saurabh
2011 arXiv   pre-print
In this paper we use several of the key ideas from Bidimensionality to give a new generic approach to design EPTASs and subexponential time parameterized algorithms for problems on classes of graphs which  ...  On the other hand, we prove that our EPTASs and subexponential time algorithms for Vertex Cover and Connected Vertex Cover carry over both to disk graphs and to unit-ball graphs in R^d for every fixed  ...  This means that the two problems neither admit EPTASs nor FPT algorithms unless FPT=W [1] , a complexity collapse considered very unlikely.  ... 
arXiv:1107.2221v2 fatcat:hc5q65lts5bl3jh5edwlx5l7w4

Faster approximation schemes and parameterized algorithms on (odd-)H-minor-free graphs

Siamak Tazari
2012 Theoretical Computer Science  
This results in a general methodology for subexponential parameterized algorithms outside the framework of bidimensionality.  ...  We obtain as a corollary that all problems with a minor-monotone subexponential kernel and amenable to our technique can be solved in subexponential FPT-time on H-minor free graphs.  ...  We showed similar results for odd-minor-free graphs and obtained the first PTAS and subexponential FPT-algorithms for Vertex Cover and Independent Set on these graph classes.  ... 
doi:10.1016/j.tcs.2011.09.014 fatcat:253avdexpzhetkbpp24762fsnm

The Complexity of Polynomial-Time Approximation

Liming Cai, Michael Fellows, David Juedes, Frances Rosamond
2007 Theory of Computing Systems  
An EPTAS is simply an FPT algorithm with respect to this parameter.  ...  An optimization problem admits a PTAS if and only if there is an algorithm Φ, that, given an input of size n, and a parameter value k = 1/ , produces a solution that is within a multiplicative factor of  ...  In §2, we introduce the necessary concepts from parameterized complexity theory and establish the connection between an EPTAS and parameterized tractability.  ... 
doi:10.1007/s00224-007-1346-y fatcat:z7x76ylombhy5i2zj5em4gt6se

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  
Erik Demaine and Moham-madTaghi Hajiaghayi survey their powerful new bidimensionality theory that studies how treewidth interacts with other structural parameters. Example 2.  ...  • The field has developed strong connections to practical computing, heuristics, and outlooks on the mission of theoretical computer science such as articulated by algorithms engineering.  ... 
doi:10.1093/comjnl/bxm111 fatcat:jhqptpkwj5c63a6t6xuqgj3dju

Algorithms for Optimization Problems in Planar Graphs (Dagstuhl Seminar 16221)

Jeff Erickson, Philip N. Klein, Dániel Marx, Claire Mathieu, Marc Herbstritt
2016 Dagstuhl Reports  
This report documents the program and the outcomes of Dagstuhl Seminar 16221 "Algorithms for Optimization Problems in Planar Graphs". The seminar was held from May 29 to June 3, 2016.  ...  This report contains abstracts for the recent developments in planar graph algorithms discussed during the seminar as well as summaries of open problems in this area of research.  ...  We show that MPT is NP-hard and quantify the connection between the two problems.  ... 
doi:10.4230/dagrep.6.5.94 dblp:journals/dagstuhl-reports/EricksonKMM16 fatcat:wasdfgivt5fqdppfxo3iqqs2ta

Parameterized complexity and approximation algorithms Dagstuhl Seminar Tree-width for rst order formulas Metakernelization: a piece of the proof Satisability Allows No Nontrivial Sparsication Unless The Polynomial-Time Hierarchy Collapses Overview of Bidimensionality

Erik Demaine, Mohammadtaghi Hajiaghayi, Daniel Marx, Erik Demaine, Mohammadtaghi Hajiaghayi, Daniel Marx, Isolde Adler, Hans Bodlaender, Holger Dell, H Berlin, Erik
unpublished
When designing approximation algorithms, we relax the requirement that the algorithm produces an optimum solution, and our aim is to devise a polynomial-time algorithm such that the solution it produces  ...  ., the Dagstuhl Seminar 09511 Parameterized complexity and approximation algorithms was held in Schloss Dagstuhl Leibniz Center for Informatics.  ...  For example, some of the most general algorithms for feedback vertex set and connected dominating set are based on bidimensionality.  ... 
fatcat:3s2jaqvz35bejfclilpaj2svrq

Parameterized Complexity and Approximation Algorithms

D. Marx
2007 Computer journal  
Approximation algorithms and parameterized complexity are usually considered to be two separate ways of dealing with hard algorithmic problems.  ...  We discuss the different ways parameterized complexity can be extended to approximation algorithms, survey results of this type, and propose directions for future research.  ...  ACKNOWLEDGMENTS I would like to thank Mike Fellows for asking me to write this survey, and for the many ideas, comments, and references that were essential in compiling the material.  ... 
doi:10.1093/comjnl/bxm048 fatcat:6yukdt7tvja3jf76zyuxfm536q

Approximation Algorithms for Unit Disk Graphs [chapter]

Erik Jan van Leeuwen
2005 Lecture Notes in Computer Science  
We prove this both indirectly by presenting a relation between unit disk graphs of bounded thickness and the pathwidth of such graphs, and directly by giving dynamic programming algorithms.  ...  We propose two new notions for unit disk graphs: thickness and density. The thickness of a graph is the number of disk centers in any width 1 slab.  ...  Hans Bodlaender for many helpful suggestions and discussions.  ... 
doi:10.1007/11604686_31 fatcat:dhf2fi67tvavhh37dog2d72bzm

Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs

Bruno Escoffier, Laurent Gourvès, Jérôme Monnot
2010 Journal of Discrete Algorithms  
In [11] , the authors propose a general technique to derive PTASs for bidimensional problems.  ...  Independently, the authors of [26,27] have also obtained FPT algorithms for MinCVC and they obtain in [27] an algorithm with complexity O * ( 2.7606 k ) .  ...  Acknowledgements We are very grateful to an anonymous referee for its insightful comments and suggestions.  ... 
doi:10.1016/j.jda.2009.01.005 fatcat:sekds3g7vvcb3hgynjppurrgge

Subexponential Parameterized Algorithms for Cut and Cycle Hitting Problems on H-Minor-Free Graphs [article]

Sayan Bandyapadhyay, William Lochet, Daniel Lokshtanov, Saket Saurabh, Jie Xue
2022 arXiv   pre-print
O(1) time algorithm for Group Feedback Edge Set and a 2^O(g √(k)log(gk))· n^O(1) time algorithm for Group Feedback Vertex Set, where g is the size of the group. 5.  ...  We design the first subexponential-time (parameterized) algorithms for several cut and cycle-hitting problems on H-minor free graphs.  ...  Methods: algorithms via a contraction decomposition theorem One of the modern tools to design polynomial-time approximation schemes (PTASs) and FPT algorithms on planar graphs or more generally on H-minor  ... 
arXiv:2111.14196v2 fatcat:2zgpjjrspjcqxm7u6xzhbneiwa

Structural Rounding: Approximation Algorithms for Graphs Near an Algorithmically Tractable Class [article]

Erik D. Demaine, Timothy D. Goodrich, Kyle Kloster, Brian Lavallee, Quanquan C. Liu, Blair D. Sullivan, Ali Vakilian, Andrew van der Poel
2018 arXiv   pre-print
Vertex Set, Minimum Maximal Matching, Chromatic Number, (ℓ-)Dominating Set, Edge (ℓ-)Dominating Set, and Connected Dominating Set.  ...  We develop a new framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common  ...  Acknowledgments We thank Abida Haque and Adam Hesterberg for helpful initial discussions, Nicole Wein for providing helpful references on bounded degeneracy problems, and Michael O'Brien for helpful comments  ... 
arXiv:1806.02771v3 fatcat:p26xojkm2jd3xcrowcx4aoxeyq

Greedy Spanners in Euclidean Spaces Admit Sublinear Separators [article]

Hung Le, Cuong Than
2021 arXiv   pre-print
We introduce a new technique that gives a simple characterization for any geometric graph to have a sublinear separator that we dub τ-lanky: a geometric graph is τ-lanky if any ball of radius r cuts at  ...  an open problem posed by Abam and Har-Peled a decade ago.  ...  Bidimensionality: New connections between FPT algorithms and PTASs.  ... 
arXiv:2107.06490v3 fatcat:rcrxzpkhlzc6fm4ssstoi3lbsq
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