New Results on a Family of Geometric Hitting Set Problems in the Plane

Abstract

We study some geometric optimal hitting set (stabbing) problems involving certain classes of objects in the plane. The objects include axis-parallel line segments, red/blue sets of pseudo-segments, axis-parallel 2-link “L” chains, pairs of line segments, etc. We examine cases in which the objects are constrained so that at least one endpoint of each object is on an inclined line (a line with slope \(-1\)). We prove that stabbing a set of vertical segments using a minimum number of horizontal segments is NP-hard when the input segments are each touching the inclined line, possibly from both sides of the line. Previously, the problem was known to be NP-hard for the general version, stabbing vertical segments with a minimum number of horizontal segments in the plane [9], and for a constrained version of this problem, in which all of the vertical segments intersect a horizontal line [3]. We provide some constant factor approximation algorithms as well. In particular, we present a PTAS for this problem using the local search technique. In contrast, if both vertical and horizontal segments are touching the inclined line from exactly one side, then the problem can be solved in polynomial time. We prove that stabbing a class of 2-link chains (“Ί-chains”) by horizontal segments is NP-hard, when both the chains and the segments have an endpoint on an inclined line and lie on one side (say, the right side) of the line. Finally, we prove that stabbing pairs of segments (each pair contains either two vertical segments or one vertical and one horizontal segments) by horizontal segments is NP-hard, when the segments are touching the inclined line from only one side.