Within the framework of pac-learning, 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 fixed-size for a class C is sufficient to ensure that the class C is pac-learnable. Previous work has shown that a class is pac-learnable if and only if the Vapnik-Chervonenkis (VC) dimension of the class i...

It is known that in general range spaces of VC-dimension 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 pseudo-disks 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 higher-dimensional spaces remain open.

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. 1

We describe a new technique for proving the existence of small ɛ-nets 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 ɛ-nets 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 pseudo-disks. This result was not known previously. We also get a very short proof for the existence of O ( 1

In this paper, we give simple and intuitive constructions to obtain linear size ɛ-nets for α-fat wedges, translations and rotations of a quadrant and axis-parallel three-sided rectangles in R 2. We also give new constructions using elementary geometry to obtain linear size weak ɛ-net for d-hypercubes and disks in R 2. 1