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### Lower bounds on stabbing lines in 3-space

M. Pellegrini
1993 Computational geometry
., 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.  ...

### Development of Algorithm for Identification of Area for Maximum Coverage and Interference

Janak Gupta, Pankaj Kumar
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.  ...

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

Haim Kaplan, Natan Rubin, Micha Sharir
2008 arXiv   pre-print
We establish a 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.  ...

### Finding stabbing lines in 3-space

M. Pellegrini, P. W. Shor
1992 Discrete & Computational Geometry
Within the same time 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]

Sepideh Aghamolaei
2021 arXiv   pre-print
of 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!  ...

### Line Transversals of Convex Polyhedra in \$\mathbb{R}^3\$

Haim Kaplan, Natan Rubin, Micha Sharir
2010 SIAM journal on computing (Print)
A 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

JAVIER CANO, FERRAN HURTADO, JORGE URRUTIA
2014 International journal of computational geometry and applications
In this paper we give 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  ...

### Minimizing the Stabbing Number of Matchings, Trees, and Triangulations

Sándor P. Fekete, Marco E. Lübbecke, Henk Meijer
2008 Discrete & Computational Geometry
The answer we provide is negative 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.  ...

### Stabbing Planes [article]

Paul Beame, Noah Fleming, Russell Impagliazzo, Antonina Kolokolova, Denis Pankratov, Toniann Pitassi, Robert Robere
2022 arXiv   pre-print
Finally, we prove linear 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.  ...

### Stabbers of line segments in the plane

Mercè Claverol, Delia Garijo, Clara I. Grima, Alberto Márquez, Carlos Seara
2011 Computational geometry
We provide efficient algorithms 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.  ...

### On Rectilinear Partitions with Minimum Stabbing Number [chapter]

Mark de Berg, Amirali Khosravi, Sander Verdonschot, Vincent van der Weele
2011 Lecture Notes in Computer Science
-There are point sets such that any partition with disjoint 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.  ...

### The Minimum Stabbing Triangulation Problem: IP Models and Computational Evaluation [chapter]

Breno Piva, Cid C. de Souza
2012 Lecture Notes in Computer Science
This paper presents integer programming (ip) formulations 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.  ...

### Lines Tangent to Four Triangles in Three-Dimensional Space

H. Bronnimann, O. Devillers, S. Lazard, F. Sottile
2007 Discrete & Computational Geometry
We investigate the 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.  ...   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.  ...

### Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

Arkadiusz Pawlik, Jakub Kozik, Tomasz Krawczyk, Michał Lasoń, Piotr Micek, William T. Trotter, Bartosz Walczak
2013 Discrete & Computational Geometry
Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-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.  ...

### The Approximability and Integrality Gap of Interval Stabbing and Independence Problems

Shalev Ben-David, Elyot Grant, Will Ma, Malcolm Sharpe
2012 Canadian Conference on Computational Geometry
Motivated by problems such as rectangle 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 )  d 2 − d + 1  Max-IS Integ.  ...
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