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Lower bounds on stabbing lines in 3-space

1993
*
Computational geometry
*

.,

doi:10.1016/0925-7721(93)90030-a
fatcat:udu4afudfndkpnmbqeuu4lshwi
*Lower**bounds*on*stabbing**lines*in 3-space, Computational Geometry: Theory and Applications 3 (1993) 53-58. ... Agarwal*for*many useful discussions. Comments of two anonymous referees have been of great help in improving the overall quality of the paper. ...*Lower**bound**for*extremal*stabbing**lines*Theorem 3. There exists a set 6%' of polyhedra in R3 of total complexity n, with Q(n") extremal*stabbing**lines*. Proof. ...##
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Development of Algorithm for Identification of Area for Maximum Coverage and Interference

2017
*
International Journal of Computer Applications
*

*For*a point set P in convex position, derive a

*lower*

*bound*on the size of the

*stabbing*set axis-parallel rectangles induced by each pair of points a,b∈P as the diagonal of the rectangles. ...

*For*a point set P, where no two points have the same x or y coordinates, derive an upper

*bound*on the size of the

*stabbing*set of axis-parallel rectangles induced by each pair of points a,b ∈ P as the ... Now we will improve the

*lower*

*bound*of

*stabbing*set,

*for*the points in general position. ...

##
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Line Transversals of Convex Polyhedra in ^3
[article]

2008
*
arXiv
*
pre-print

We establish a

arXiv:0807.1221v1
fatcat:jsu4fcl2xngbbauvxwn4kf5mbi
*bound*of O(n^2k^1+),*for*any >0, on the combinatorial complexity of the set of*line*transversals of a collection of k convex polyhedra in ^3 with a total of n facets, and present a randomized ... To obtain the above result, we study the set of*line*transversals which emanate from a fixed*line*ℓ_0, establish an almost tight*bound*of O(nk^1+) on the complexity of , and provide a randomized algorithm ...*For*the sake of simplicity, we derive the above*bound*only*for*extremal*stabbing**lines*defined by pairs of polyhedra. ...##
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Finding stabbing lines in 3-space

1992
*
Discrete & Computational Geometry
*

Within the same time

doi:10.1007/bf02293043
fatcat:xo2rvmyj45flxc2e3546zmniky
*bound*it is possible to determine if a*stabbing**line*exists and to nd one. ... A*line*intersecting all polyhedra in a set B is called a \stabber"*for*the set B. This paper addresses some combinatorial and algorithmic questions about the set S(B) of all*lines**stabbing*B. ... Acknowledgments We wish to thank Micha Sharir*for*proposing the problem and Richard Pollack, Boris Aronov, Pankaj K. Agarwal and Janos Pach*for*many useful discussions. ...##
###
Catching a Polygonal Fish with a Minimum Net
[article]

2021
*
arXiv
*
pre-print

of

arXiv:2008.06337v3
fatcat:7k3bengatrhsfjnwe4jehmshx4
*lines*is minimized. ... We prove the solution is always a regular grid or a set of equidistant parallel*lines*, whose distance depends on P. ... Such a net must*stab*any subset of copies of a shape, and the minimum number of*lines**for**stabbing*all copies is a*lower**bound*on the minimum number of*lines*in the minimum net. Catching the Fish! ...##
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Line Transversals of Convex Polyhedra in $\mathbb{R}^3$

2010
*
SIAM journal on computing (Print)
*

A

doi:10.1137/080744694
fatcat:4ei42ypusjd33cq7z6kywkwv3y
*line*is a transversal of P if it intersects every member of P. The set of all*line*transversals of P is called the transversal space (or*stabbing*region) of P and is denoted by T (P). ... We establish a*bound*of O(n 2 k 1+ε ),*for*any ε > 0, on the combinatorial complexity of the set T of*line*transversals of a collection P of k convex polyhedra in R 3 with a total of n facets, and we present ... We thank the anonymous referees*for*valuable suggestions that helped us to improve the presentation. ...##
###
STABBING SIMPLICES OF POINT SETS WITH k-FLATS

2014
*
International journal of computational geometry and applications
*

In this paper we give

doi:10.1142/s021819591460005x
fatcat:h3hb7uvdtrbqphf7xnzxfpc7y4
*lower*and upper*bounds*on the size of minimum m k -stabbers of point sets in R d . We study mainly m k -stabbers in the plane and in R 3 . ... In our previous terminology, determine*lower*and upper*bounds**for*f 1 1 (n) and f 2 1 (n)*for*point sets on the plane. ... The*lower**bound*follows from the fact that r*lines*, no three of which intersect at a point, divide the plane into 2 + 2 + 3 + · · · + r convex regions, and if a set of r*lines**stabs*all of the triangles ...##
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Minimizing the Stabbing Number of Matchings, Trees, and Triangulations

2008
*
Discrete & Computational Geometry
*

The answer we provide is negative

doi:10.1007/s00454-008-9114-6
fatcat:lztesrasybbsnnk2yo5s7bgaq4
*for*a number of minimum*stabbing*problems by showing them NP-hard by means of a general proof technique. It implies non-trivial*lower**bounds*on the approximability. ... The (axis-parallel)*stabbing*number of a given set of*line*segments is the maximum number of segments that can be intersected by any one (axis-parallel)*line*. ... We also thank Kamal Jain*for*some discussions on iterated rounding. ...##
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Stabbing Planes
[article]

2022
*
arXiv
*
pre-print

Finally, we prove linear

arXiv:1710.03219v2
fatcat:4dedtl7iordyvcmquixcpwmd6y
*lower**bounds*on the rank of*Stabbing*Planes refutations by adapting*lower**bounds*in communication complexity and use these*bounds*in order to show that*Stabbing*Planes proofs cannot ... We develop a new semi-algebraic proof system called*Stabbing*Planes which formalizes modern branch-and-cut algorithms*for*integer programming and is in the style of DPLL-based modern SAT solvers. ... n)*lower**bound**for*both formulas. ...##
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Stabbers of line segments in the plane

2011
*
Computational geometry
*

We provide efficient algorithms

doi:10.1016/j.comgeo.2010.12.004
fatcat:ogmalirgnjhv3hfb5v7privnbu
*for*the following problems: computing the*stabbing*wedges*for*S, finding a*stabbing*wedge*for*a set of parallel segments with equal length, and computing other stabbers ... The problem of computing a representation of the*stabbing**lines*of a set S of segments in the plane was solved by Edelsbrunner et al. ... Hurtado*for*their help in the first study of the*stabbing*problems which was part of her PhD thesis. E. Arkin and J. S. B. Mitchell deserve our thanks*for*their useful comments. B. ...##
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On Rectilinear Partitions with Minimum Stabbing Number
[chapter]

2011
*
Lecture Notes in Computer Science
*

-There are point sets such that any partition with disjoint

doi:10.1007/978-3-642-22300-6_26
fatcat:tysdeipp5fcvthhhxev4vkakmq
*bounding*boxes has*stabbing*number Ω(r 1−1/d ), while the optimal partition has*stabbing*number 2. ... A rectilinear r-partition*for*S is a collection Ψ (S) := {(S1, b1), . . . , (St, bt)}, such that the sets Si form a partition of S, each bi is the*bounding*box of Si, and n/2r |Si| 2n/r*for*all 1 i t. ... The arcs from the boxes to the sink also have (besides the upper*bound*of 2n/r on the flow) a*lower**bound*of n/2r on the flow. ...##
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The Minimum Stabbing Triangulation Problem: IP Models and Computational Evaluation
[chapter]

2012
*
Lecture Notes in Computer Science
*

This paper presents integer programming (ip) formulations

doi:10.1007/978-3-642-32147-4_5
fatcat:syy6jyqnyneqfoqkcpdnv5d3mm
*for*the mstr, that allow us to solve it exactly through ip branch-and-*bound*(b&b) algorithms. ... The minimum*stabbing*triangulation of a set of points in the plane (mstr) was previously investigated in the literature. ... Lübbecke*for*making available the grid instances. ...##
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Lines Tangent to Four Triangles in Three-Dimensional Space

2007
*
Discrete & Computational Geometry
*

We investigate the

doi:10.1007/s00454-006-1278-3
fatcat:vc5udpwucvbg5gu67siwj4bnc4
*lines*tangent to four triangles in R 3 . By a construction, there can be as many as 62 tangents. ... In this paper we consider the case of four triangles in R 3 , and establish*lower*and upper*bounds*on the number of tangent*lines*. ... [7] showed an (n 3 )*lower**bound*on the complexity of the set of free*lines*(and thus free segments) among n disjoint homothetic convex polyhedra. Recently, Agarwal et al. ...##
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Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

2013
*
Discrete & Computational Geometry
*

Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-

doi:10.1007/s00454-013-9534-9
fatcat:sccdix27orhd5p2ucvcdfchhvq
*line*coloring of intervals on the*line*. ... This provides a negative answer to a question of Gyarfas and Lehel*for*L-shapes. ... of*line*segments is*bounded*by an absolute constant. ...##
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The Approximability and Integrality Gap of Interval Stabbing and Independence Problems

2012
*
Canadian Conference on Computational Geometry
*

Motivated by problems such as rectangle

dblp:conf/cccg/Ben-DavidGMS12
fatcat:bqo6cp7ol5ebzm7nlpjbbwz2j4
*stabbing*in the plane, we study the minimum hitting set and maximum independent set problems*for*families of d-intervals and d-union-intervals. ...*for*the hitting set problem on d-intervals; (3) a proof that the approximation ratios*for*independent set on families of 2-intervals and 2-union-intervals can be improved to match tight duality gap*lower*... The table below summarizes the known integrality and duality gap*bounds**for*large d: d-Interval*Lower**Bound*Upper*Bound*Duality Gap Ω( d 2 log d ) [13] d 2 − d + 1 [10] Max-IS Integ. ...
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