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...

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 ...

We consider a generalization of the set cover problem, in which elements are covered by pairs of objects, and we are required to find a minimum cost subset of objects that induces a collection of pairs covering all elements. Formally, let U be a ground set of elements and let S be a set of objects, where each object i has a non-negative cost wi. For every {i, j} ⊆ S, let C(i, j) be the collecti...

We consider the minimum entropy principle for learning data generated by a random source and observed with random noise. In our setting we have a sequence of observations of objects drawn uniformly at random from a population. Each object in the population belongs to one class. We perform an observation for each object which determines that it belongs to one of a given set of classes. Given the...

We consider the problem of finding a minimum identifying code in a graph, i.e., a designated set of vertices whose neighborhoods uniquely overlap at any vertex on the graph. This identifying code problem was initially introduced in 1998 and has been since fundamentally connected to a wide range of applications, including fault diagnosis, location detection, environmental monitoring, and connect...

We study minimum entropy submodular set cover, a variant of the submodular set cover problem (Wolsey [22], Fujito [11], etc) that generalizes the minimum entropy set cover problem (Halperin and Karp [12], Cardinal et al. [5]) We give a general bound of the approximation performance of the greedy algorithm using an approach that can be interpreted in terms of a particular type of biased network ...