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The Parameterized Complexity of Some Geometric Problems in Unbounded Dimension
[chapter]
2009
Lecture Notes in Computer Science
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We study the parameterized complexity of the following fundamental geometric problems with respect to the dimension d: i) Given n points in , compute their minimum enclosing cylinder. ii) Given two n-point ...
We show that (the decision versions of) all these problems are W[1]-hard when parameterized by the dimension d. in O(f(d)n^c) time, for any computable function f and constant c (under the Exponential Time ...
The dimension of geometric problems is a natural parameter for studying their parameterized complexity. ...
doi:10.1007/978-3-642-11269-0_16
fatcat:7bukjj2g7re27eywrz6zcy2ocq
On the Parameterized Complexity of d-Dimensional Point Set Pattern Matching
[chapter]
2006
Lecture Notes in Computer Science
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When the dimension d is unbounded, the problem is equivalent to graph isomorphism and is conjectured to be in FPT. ...
Deciding whether two n-point sets A, B ∈ R d are congruent is a fundamental problem in geometric pattern matching. ...
Parameterized complexity theory measures the complexity of hard algorithmic problems in terms of parameters in addition to the problem input size. ...
doi:10.1007/11847250_16
fatcat:ftfickzctbccjdkdgxkf2hjmcy
On the parameterized complexity of d-dimensional point set pattern matching
2008
Information Processing Letters
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When the dimension d is unbounded, the problem is equivalent to graph isomorphism and is conjectured to be in FPT. ...
Deciding whether two n-point sets A, B ∈ R d are congruent is a fundamental problem in geometric pattern matching. ...
Parameterized complexity theory measures the complexity of hard algorithmic problems in terms of parameters in addition to the problem input size. ...
doi:10.1016/j.ipl.2007.08.003
fatcat:622yn35rrzgl3h6sucheftaugy
Dirac operators and spectral triples for some fractal sets built on curves
[article]
2007
arXiv
pre-print
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In these cases, we show that our spectral triples do describe the geodesic distance and the Minkowski dimension as well as, more generally, the complex fractal dimensions of the space. ...
Furthermore, in the case of the Sierpinski gasket, the associated Dixmier-type trace coincides with the normalized Hausdorff measure of dimension 3/ 2. ...
This is in agreement with a conjecture made in [28] , Section 8, when discussing the 'geometric complex dimensions' of the gasket. ...
arXiv:math/0610222v2
fatcat:cetfc2e47jd7hn7k2lwvyjzp2i
Dirac operators and spectral triples for some fractal sets built on curves
2008
Advances in Mathematics
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In these cases, we show that our spectral triples do describe the geodesic distance and the Minkowski dimension as well as, more generally, the complex fractal dimensions of the space. ...
Furthermore, in the case of the Sierpinski gasket, the associated Dixmier-type trace coincides with the normalized Hausdorff measure of dimension log 3/ log 2. ...
Acknowledgments We are grateful to Alexander Teplyaev for constructive discussions about analysis on fractals and for bringing the newly developed theory of 'quantum graphs' to our attention, after we ...
doi:10.1016/j.aim.2007.06.009
fatcat:bjr5w47iovaszi53w4z2n2zfu4
Bounding the Vapnik-Chervonenkis dimension of concept classes parameterized by real numbers
1995
Machine Learning
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The Vapnik-Chervonenkis (V-C) dimension is an important combinatorial tool in the analysis of learning problems in the PAC framework. ...
For polynomial learnability, we seek upper bounds on the V-C dimension that are polynomial in the syntactic complexity of concepts. ...
Blumer et al. (1987) formalize this notion in their definition of "Occam algorithm". ...
doi:10.1007/bf00993408
fatcat:24i563dbnzap3hro4bggt65myy
Parameterized Complexity and Approximation Algorithms
2007
Computer journal
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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
Duality and Geometry Straightness, Characterization and Envelope
[chapter]
2006
Lecture Notes in Computer Science
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In this paper, we focus on two kinds of duality/polarity applied to geometrical problems: digital straightness detection and envelope computation. ...
Duality applied to geometrical problems is widely used in many applications in computer vision or computational geometry. ...
Indeed, many complex geometrical concepts are used and the overall algorithm is not really tractable. ...
doi:10.1007/11907350_1
fatcat:epyomrm5vnaqjny6jpdla5lbfy
Parameterized Complexity of Geometric Problems
2007
Computer journal
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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
Quartic Supercyclides for Geometric Design
[chapter]
2002
From Geometric Modeling to Shape Modeling
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This leads to significant problems in certain types of geometric computation, and also in the exchange of CAD models between different systems. ...
The paper surveys the current state of the art, and indicates some fruitful areas for future work. ...
This requires that algorithms are efficient in handling two types of complexity: • global complexity -the existence in an intersection curve of numerous disjoint branches, some of which can be small closed ...
doi:10.1007/978-0-387-35495-8_15
fatcat:friehypxbzbj7b6ml4ic3gnzca
Page 7717 of Mathematical Reviews Vol. , Issue 98M
[page]
1998
Mathematical Reviews
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In their proofs they make use of L?-techniques for the 0-complex, the ordinary dimension, dimc, of cohomology spaces being replaced by the von Neumann [-dimension, dimr. ...
Through any given point there is a biholomorphically invariant family of two-(real)- dimensional surfaces in M, parameterized by C’, called chains. ...
The parameterized complexity of finding a 2-sphere in a simplicial complex
[article]
2018
arXiv
pre-print
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We consider the problem of finding a subcomplex K' of a simplicial complex K such that K' is homeomorphic to the 2-dimensional sphere, S^2. We study two variants of this problem. ...
The second problem is the dual of the first, and asks if K' can be found by removing at most k triangles from K. ...
We are grateful to the other participants of the workshop and the Lorentz Center for their support. ...
arXiv:1802.07175v1
fatcat:3e2zlnzkovbopbzhdr5qprr3fy
FO model checking on geometric graphs
2019
Computational geometry
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We study the FO model checking problem for dense graph classes definable by geometric means (intersection and visibility graphs). ...
Over the past two decades the main focus of research into first-order (FO) model checking algorithms has been on sparse relational structures -culminating in the FPT algorithm by Grohe, Kreutzer and Siebertz ...
parameterized complexity. ...
doi:10.1016/j.comgeo.2018.10.001
fatcat:4reekwyaqfez7idoh34lgyksoe
Experience in the exchange of procedural shape models using ISO 10303 (STEP)
2006
Proceedings of the 2006 ACM symposium on Solid and physical modeling - SPM '06
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This paper reports in some detail on one of those trials, and comments on the experience gained. ...
The transfer of parameterized assembly models is an additional objective. ...
In STEP terms this may be an unbounded line defined by a point and a direction, or some equivalent construct. ...
doi:10.1145/1128888.1128920
dblp:conf/sma/PrattK06
fatcat:otggjcfvkzfdlii7zhldpijhci
ON PARAMETERIZED COMPLEXITY OF HITTING SET PROBLEM FOR AXIS–PARALLEL SQUARES INSTERSECTING A STRAIGHT LINE
2016
Ural Mathematical Journal
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The Hitting Set Problem (HSP) is the well known extremal problem adopting research interest in the fields of combinatorial optimization, computational geometry, and statistical learning theory for decades ...
In the general setting, the problem is NP-hard and hardly approximable. ...
Introduction We consider the parameterized complexity of a geometric statement of the well-known Hitting Set Problem (HSP), engaging researchers in combinatorial optimization, computational geometry and ...
doi:10.15826/umj.2016.2.010
fatcat:dvw6jywqljhq3b777p2r6e7bmu
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