Approximability and hardness of geometric covering and hitting problems

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.