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Planar Support for Non-piercing Regions and Applications

Govindarajan, Rajiv Raman, and Saurabh Ray . Packing and covering with non-piercing regions . Discrete & Computational Geometry , Mar 2018 . doi: 10 .1007/s00454-018-9983-2. On planar supports for hypergraphs ... Computational Geometry , 2018 . URL: http: //arxiv.org/abs/1711.05473. Nabil H. Mustafa , Rajiv Raman, and Saurabh Ray . Quasi-polynomial time approximation scheme for weighted geometric set cover on pseudodisks

On a Problem of Danzer

Let C be a bounded convex object in R^d, and P a set of n points lying outside C. Further let c_p, c_q be two integers with 1 <= c_q <= c_p <= n - floor[d/2], such that every c_p + floor[d/2] points of P contains a subset of size c_q + floor[d/2] whose convex-hull is disjoint from C. Then our main theorem states the existence of a partition of P into a small number of subsets...

Near-Optimal Generalisations of a Theorem of Macbeath

The existence of Macbeath regions is a classical theorem in convex geometry ("A Theorem on non-homogeneous lattices", Annals of Math, 1952). We refer the reader to the survey of I. Barany for several applications. Recently there have been some striking applications of Macbeath regions in discrete and computational geometry. In this paper, we study Macbeath's problem in a more...

Improved Results on Geometric Hitting Set Problems

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...