Geometric data summarization has become an essential tool in both geometric
approximation algorithms and where geometry intersects with big data problems. In linear or …

Unsolved problems in visibility graphs of points, segments, and polygons Page 1 22 Unsolved
Problems in Visibility Graphs of Points, Segments, and Polygons SUBIR K. GHOSH, Tata …

The massive volume of data generated in modern applications can overwhelm our ability to
conveniently transmit, store, and index it. For many scenarios, building a compact summary …

The minimum-weight set cover problem is widely known to be O (log n)-approximable, with
no improvement possible in the general case. We take the approach of exploiting problem …

We investigate the family of intersection graphs of low density objects in low dimensional
Euclidean space. This family is quite general, includes planar graphs, and in particular is a …

There has been much progress on geometric set cover problems, but most known
techniques only apply to the unweighted setting. For the weighted setting, very few results …

We show that for any sequence $ f:{\bf N}\to\{-1,+ 1\} $ taking values in $\{-1,+ 1\} $, the
discrepancy $$\sup_ {n, d\in {\bf N}}\left|\sum_ {j= 1}^ nf (jd)\right| $$ of $ f $ is infinite. This …

We study several geometric set cover and set packing problems involving configurations of
points and geometric objects in Euclidean space. We show that it is APX-hard to compute a …

According to a well known theorem of Haussler and Welzl (1987), any range space of
bounded VC-dimension admits an ε-net of size O (1/ε log 1/ε). Using probabilistic …

The use of random samples to approximate properties of geometric configurations has been
an influential idea for both combinatorial and algorithmic purposes. This chapter considers …