Hitting sets when the VC-dimension is small

Even, Guy; Rawitz, Dror; Shahar, Shimon

doi:10.1016/j.ipl.2005.03.010

Cited by 93 publications

(85 citation statements)

References 12 publications

(21 reference statements)

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“…If we can compute an ǫ-net of size c/ǫ for the weighted ǫ-net problem for R in polynomial time then we can compute a hitting set of size at most c · OPT for R, where OPT is the size of the optimal (smallest) hitting set, in polynomial time. A shorter, simpler proof was given by Even et al [5].…”

Section: Introductionmentioning

confidence: 91%

Improved Results on Geometric Hitting Set Problems

Mustafa

Ray

2010

Discrete Comput Geom

164

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We consider the problem of computing minimum geometric hitting sets in which, given a set of geometric objects and a set of points, the goal is to compute the smallest subset of points that hit all geometric objects. The problem is known to be strongly NP-hard even for simple geometric objects like unit disks in the plane. Therefore, unless P=NP, it is not possible to get Fully Polynomial Time Approximation Algorithms (FPTAS) for such problems. We give the first PTAS for this problem when the geometric objects are half-spaces in R 3 and when they are an r-admissible set regions in the plane (this includes pseudo-disks as they are 2-admissible). Quite surprisingly, our algorithm is a very simple local search algorithm which iterates over local improvements only.

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Section: Introductionmentioning

confidence: 91%

Improved Results on Geometric Hitting Set Problems

Mustafa

Ray

2010

Discrete Comput Geom

164

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“…This observation has far reaching consequences. The construction of small epsilon-nets has become one of the most powerful general techniques in computational geometry (see [Ch00,EvRS05]). …”

Section: Introductionmentioning

confidence: 99%

Tight lower bounds for the size of epsilon-nets

Pach¹,

Tardos²

2012

J. Amer. Math. Soc.

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According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VC-dimension admits an ε-net of size O 1 ε log 1 ε . Using probabilistic techniques, Pach and Woeginger (1990) showed that there exist range spaces of VCdimension 2, for which the above bound is sharp. The only known range spaces of small VC-dimension, in which the ranges are geometric objects in some Euclidean space and the size of the smallest ε-nets is superlinear in 1 ε , were found by Alon (2010). In his examples, the size of the smallest ε-nets is Ω 1 ε g( 1 ε ) , where g is an extremely slowly growing function, closely related to the inverse Ackermann function.We show that there exist geometrically defined range spaces, already of VCdimension 2, in which the size of the smallest ε-nets is Ω 1 ε log 1 ε . We also construct range spaces induced by axis-parallel rectangles in the plane, in which the size of the smallest ε-nets is Ω 1 ε log log 1 ε . By a theorem of Aronov, Ezra, and Sharir (2010), this bound is tight.

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“…In particular, Brönnimann and Goodrich [12] have proposed an almost optimal solution to the disk cover algorithm, i.e., to finding a minimum number of disks in a given family that cover a given set of points. Theorem 2 allows one to extend this result to arbitrary Bregman ball cover (see also [21]). …”

Section: Theorem 2 the Vc-dimension Of The Class Of All Bregman Ballsmentioning

confidence: 94%

Bregman Voronoi Diagrams

Boissonnat

Nielsen

Nock³

2010

Discrete Comput Geom

126

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The Voronoi diagram of a finite set of objects is a fundamental geometric structure that subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define various variants of Voronoi diagrams depending on the class of objects, the distance function and the embedding space. In this paper, we investigate a framework for defining and building Voronoi diagrams for a broad class of distance functions called Bregman divergences. Bregman divergences include not only the traditional (squared) Euclidean distance but also various divergence measures based on entropic functions. Accordingly, Bregman Voronoi diagrams allow one to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We define several types of Bregman diagrams, establish correspondences between those diagrams (using the Legendre transformation), and show how to compute them efficiently. We also introduce extensions of these diagrams, e.g., k-order and k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set of points and their connection with Bregman Voronoi diagrams. We show that these triangulations capture many of the properties of the celebrated Delaunay triangulation.

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Hitting sets when the VC-dimension is small

Cited by 93 publications

References 12 publications

Improved Results on Geometric Hitting Set Problems

Improved Results on Geometric Hitting Set Problems

Tight lower bounds for the size of epsilon-nets

Bregman Voronoi Diagrams

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