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The Parameterized Complexity of Some Geometric Problems in Unbounded Dimension
[chapter]

2009
*
Lecture Notes in Computer Science
*

We study

doi:10.1007/978-3-642-11269-0_16
fatcat:7bukjj2g7re27eywrz6zcy2ocq
*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*. ...##
###
On the Parameterized Complexity of d-Dimensional Point Set Pattern Matching
[chapter]

2006
*
Lecture Notes in Computer Science
*

When

doi:10.1007/11847250_16
fatcat:ftfickzctbccjdkdgxkf2hjmcy
*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. ...##
###
On the parameterized complexity of d-dimensional point set pattern matching

2008
*
Information Processing Letters
*

When

doi:10.1016/j.ipl.2007.08.003
fatcat:622yn35rrzgl3h6sucheftaugy
*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. ...##
###
Dirac operators and spectral triples for some fractal sets built on curves
[article]

2007
*
arXiv
*
pre-print

*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. ...

##
###
Dirac operators and spectral triples for some fractal sets built on curves

2008
*
Advances in Mathematics
*

*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 ...

##
###
Bounding the Vapnik-Chervonenkis dimension of concept classes parameterized by real numbers

1995
*
Machine Learning
*

*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". ...

##
###
Parameterized Complexity and Approximation Algorithms

2007
*
Computer journal
*

Approximation algorithms and

doi:10.1093/comjnl/bxm048
fatcat:6yukdt7tvja3jf76zyuxfm536q
*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. ...##
###
Duality and Geometry Straightness, Characterization and Envelope
[chapter]

2006
*
Lecture Notes in Computer Science
*

*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. ...

##
###
Parameterized Complexity of Geometric Problems

2007
*
Computer journal
*

This paper surveys

doi:10.1093/comjnl/bxm053
fatcat:ohzdxo2ehbcgrawsgmzdcvz5xq
*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. ...##
###
Quartic Supercyclides for Geometric Design
[chapter]

2002
*
From Geometric Modeling to Shape Modeling
*

This leads to significant

doi:10.1007/978-0-387-35495-8_15
fatcat:friehypxbzbj7b6ml4ic3gnzca
*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 ...##
###
Page 7717 of Mathematical Reviews Vol. , Issue 98M
[page]

1998
*
Mathematical Reviews
*

*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

We consider

arXiv:1802.07175v1
fatcat:3e2zlnzkovbopbzhdr5qprr3fy
*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. ...##
###
FO model checking on geometric graphs

2019
*
Computational geometry
*

We study

doi:10.1016/j.comgeo.2018.10.001
fatcat:4reekwyaqfez7idoh34lgyksoe
*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*. ...##
###
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
*

This paper reports

doi:10.1145/1128888.1128920
dblp:conf/sma/PrattK06
fatcat:otggjcfvkzfdlii7zhldpijhci
*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. ...##
###
ON PARAMETERIZED COMPLEXITY OF HITTING SET PROBLEM FOR AXIS–PARALLEL SQUARES INSTERSECTING A STRAIGHT LINE

2016
*
Ural Mathematical Journal
*

*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 ...

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