We consider various hitting and piercing problems for the family of axis-parallel rectangles induced by a point set. Selection Lemmas on induced objects are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon nets and slimming Delaunay triangulations. Selection Lemma type results typically bound the maximum number...

A common subproblem of $DNF$ approximate counting and derandomizing $RL$ is the discrepancy problem for combinatorial rectangles. We explicitly construct a $poly(n)$size sample space that approximates the volume of any combinatorial rectangle in $[n]^n$ to within $o(1)$ error (improving on the constructions of [EGLNV92]). The construction extends the techniques of [LLSZ95] for the analogous hit...

The problem of finding an optimal vertex cover in a graph is a classic NP-complete problem, and is a special case of the hitting set question. On the other hand, the hitting set problem, when asked in the context of induced geometric objects, often turns out to be exactly the vertex cover problem on restricted classes of graphs. In this work we explore a particular instance of such a phenomenon...

EDITOR’S NOTE Geopersia is published biannually by the College of Science, University of Tehran, Iran. The College of Science published “Journal of Science”, which covered all fields of basic science, from 1968 to 2010. It was then decided that the College’s journals should be more specialized, and hence publication of Geopersia was launched. Geopersia aims to publish peer-reviewed original...

We introduce the successive Hitting Set model, where the set system is not given in advance but a set generator produces the sets that contain a specific element from the universe on demand. Despite incomplete knowledge about the set system, we show that several approximation algorithms for the conventional Hitting Set problem can be adopted to perform well in this model. We describe, and exper...

We study a geometric hitting set problem involving unirectional rays and curves in the plane. We show that this problem is hard to approximate within a logarithmic factor even when the curves are convex polygonal x-monotone chains. Additionally, it is hard to approximate within a factor of 76 even when the curves are line segments with bounded slopes. Lastly, we demonstrate that the problem is ...

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Ice-core records show that climate changes in the past have been large, rapid, and synchronous over broad areas extending into low latitudes, with less variability over historical times. These ice-core records come from high mountain glaciers and the polar regions, including small ice caps and the large ice sheets of Greenland and Antarctica.