Constrained Orthogonal Segment Stabbing

TL;DR

This paper investigates a constrained version of the orthogonal segment stabbing problem, proving NP-hardness while also developing constant approximation algorithms under specific input restrictions.

Contribution

It introduces a constrained problem variant with a common vertical line intersection, establishing NP-hardness and providing constant approximation algorithms for certain cases.

Findings

Problem remains NP-hard under constraints

Constant approximation algorithms are achievable

Multiple problem variants analyzed

Abstract

Let $S$ and $D$ each be a set of orthogonal line segments in the plane. A line segment $s\in S$ \emph{stabs} a line segment $s'\in D$ if $s\cap s'\neq\emptyset$. It is known that the problem of stabbing the line segments in $D$ with the minimum number of line segments of $S$ is NP-hard. However, no better than $O(\log |S\cup D|)$-approximation is known for the problem. In this paper, we introduce a constrained version of this problem in which every horizontal line segment of $S\cup D$ intersects a common vertical line. We study several versions of the problem, depending on which line segments are used for stabbing and which line segments must be stabbed. We obtain several NP-hardness and constant approximation results for these versions. Our finding implies, the problem remains NP-hard even under the extra assumption on input, but small constant approximation algorithms can be designed.