Results 1 
7 of
7
Sample compression, learnability, and the VapnikChervonenkis dimension
 MACHINE LEARNING
, 1995
"... Within the framework of paclearning, we explore the learnability of concepts from samples using the paradigm of sample compression schemes. A sample compression scheme of size k for a concept class C ` 2 X consists of a compression function and a reconstruction function. The compression function r ..."
Abstract

Cited by 61 (3 self)
 Add to MetaCart
Within the framework of paclearning, we explore the learnability of concepts from samples using the paradigm of sample compression schemes. A sample compression scheme of size k for a concept class C ` 2 X consists of a compression function and a reconstruction function. The compression function receives a finite sample set consistent with some concept in C and chooses a subset of k examples as the compression set. The reconstruction function forms a hypothesis on X from a compression set of k examples. For any sample set of a concept in C the compression set produced by the compression function must lead to a hypothesis consistent with the whole original sample set when it is fed to the reconstruction function. We demonstrate that the existence of a sample compression scheme of fixedsize for a class C is sufficient to ensure that the class C is paclearnable. Previous work has shown that a class is paclearnable if and only if the VapnikChervonenkis (VC) dimension of the class i...
How to Net a Lot with Little: Small εNets for Disks and Halfspaces
 In Proc. 6th Annu. ACM Sympos. Comput. Geom
, 2000
"... It is known that in general range spaces of VCdimension d ? 1 require "nets to be of size at least\Omega\Gamma " ). We address the question whether this general lower bound is valid for the special range spaces that typically arise in computational geometry. We show that disks and pseudodi ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
It is known that in general range spaces of VCdimension d ? 1 require "nets to be of size at least\Omega\Gamma " ). We address the question whether this general lower bound is valid for the special range spaces that typically arise in computational geometry. We show that disks and pseudodisks in the plane as well as halfspaces in R allow "nets of size only O( " ), which is best possible up to a multiplicative constant. The analogous questions for higherdimensional spaces remain open.
PTAS for geometric hitting set problems via local search
 In Symposium on Computational Geometry
, 2009
"... 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 NPhard even for simple geometric objects like ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
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 NPhard 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 halfspaces in R 3 and when they are an radmissible set regions in the plane (this includes pseudodisks as they are 2admissible). Quite surprisingly, our algorithm is a very simple local search algorithm which iterates over local improvements only.
Improved results on geometric hitting set problems
 In Proceedings of Symposium on Computational Geometry
, 2009
"... 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 NPhard even for simple geometric objects like ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
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 NPhard 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 halfspaces in R 3 and when they are an radmissible set regions in the plane (this includes pseudodisks as they are 2admissible). Quite surprisingly, our algorithm is a very simple local search algorithm which iterates over local improvements only. 1
unknown title
"... We describe a new technique for proving the existence of small ɛnets for hypergraphs satisfying certain simple conditions. The technique is particularly useful for proving o ( 1 1 ɛ log ɛ) upper bounds which is not possible using the standard VC dimension theory. We apply the technique to several g ..."
Abstract
 Add to MetaCart
We describe a new technique for proving the existence of small ɛnets for hypergraphs satisfying certain simple conditions. The technique is particularly useful for proving o ( 1 1 ɛ log ɛ) upper bounds which is not possible using the standard VC dimension theory. We apply the technique to several geometric hypergraphs and obtain simple proofs for the existence of O ( 1 ɛ) size ɛnets for them. This includes the geometric hypergraph in which the vertex set is a set of points in the plane and the hyperedges are defined by a set of pseudodisks. This result was not known previously. We also get a very short proof for the existence of O ( 1
New ɛNet Constructions
"... In this paper, we give simple and intuitive constructions to obtain linear size ɛnets for αfat wedges, translations and rotations of a quadrant and axisparallel threesided rectangles in R 2. We also give new constructions using elementary geometry to obtain linear size weak ɛnet for dhypercube ..."
Abstract
 Add to MetaCart
In this paper, we give simple and intuitive constructions to obtain linear size ɛnets for αfat wedges, translations and rotations of a quadrant and axisparallel threesided rectangles in R 2. We also give new constructions using elementary geometry to obtain linear size weak ɛnet for dhypercubes and disks in R 2. 1
GEOMETRIC SET COVER AND HITTING SETS FOR POLYTOPES IN R 3 SÖREN LAUE
, 2008
"... Abstract. Suppose we are given a finite set of points P in R 3 and a collection of polytopes T that are all translates of the same polytope T. We consider two problems in this paper. The first is the set cover problem where we want to select a minimal number of polytopes from the collection T such t ..."
Abstract
 Add to MetaCart
Abstract. Suppose we are given a finite set of points P in R 3 and a collection of polytopes T that are all translates of the same polytope T. We consider two problems in this paper. The first is the set cover problem where we want to select a minimal number of polytopes from the collection T such that their union covers all input points P. The second problem that we consider is finding a hitting set for the set of polytopes T, that is, we want to select a minimal number of points from the input points P such that every given polytope is hit by at least one point. We give the first constantfactor approximation algorithms for both problems. We achieve this by providing an epsilonnet for translates of a polytope in R 3 of size O ( 1 ǫ).