### Abstract:

In this thesis, we study the following three geometric covering and hitting problems:
(i) red-blue set cover with unit disks, (ii) set cover and hitting set problems with axisparallel rectangles and points in the plane, and (iii) stabbing line segments with disks and
related problems.
Given a set S of unit disks, a set R of red points, and a set B of blue points, the
red-blue set cover problem with unit disks asks us to find a subset S
0 ⊆ S covering all
blue points in B such that S
0
covers the minimum number of red points from R. In
this thesis, we give the first constant factor approximation algorithm for this problem.
The algorithm consists of three steps. First, we give a polynomial-time algorithm for
the line-separable red-blue set cover problem. Then, we give a 2-factor approximation
algorithm for the strip-separable red-blue set cover problem and finally we combine these
results with the results of Ambuhl et al. [ ¨ 4] to obtain a constant factor approximation
algorithm for red-blue set cover with unit disks. Our polynomial-time algorithm for the
line-separable red-blue set cover problem involves a novel decomposition of the optimal
solution into blocks with unique structure and extensions of the sweep-line technique of
Erlebach and van Leeuwen [23].
Next, we study covering and hitting problems with axis-parallel rectangles in the
plane. We consider the following two scenarios: (a) all rectangles share a common point
and (b) the side lengths of each rectangle are integers bounded by a constant K. For the
first case, we show that both set cover and hitting set problems are NP-hard with two
types of rectangles and hitting set problem is APX-hard with arbitrary number of rectangle types. For rectangles with bounded integer side lengths, we give PTASes for both
set cover and hitting set problems. On the other hand, we show that both problems are
NP-hard, even with 1 × 2 and 2 × 1 rectangles, when all rectangles intersect a unit-height
horizontal strip.
Finally, we consider stabbing problems in the plane where we need to stab a set X
of objects with a minimum size subset of a set Y of objects. We show that stabbing axis-parallel line segments with unit disks and stabbing unit disks with axis-parallel line
segments are both APX-hard. However, in the special case when no two line segments
intersect, we present PTASes for both problems. Finally, we show that stabbing circles
with unit circles is APX-hard, while stabbing disks with disks admits a PTAS.