Stabbing line segments with disks: complexity and approximation algorithms

TL;DR

This paper investigates the computational complexity of covering line segments in planar graphs with fixed-radius disks, proving NP-hardness in many cases and proposing efficient approximation algorithms under certain geometric constraints.

Contribution

It establishes NP-hardness results for the problem on various graph classes and introduces a fast approximation algorithm for specific geometric conditions.

Findings

NP-hardness for Delaunay triangulations and Gabriel graphs

A fast O(|E| log |E|) approximation algorithm

Effective when edge lengths are uniformly bounded by the disk radius

Abstract

Computational complexity and approximation algorithms are reported for a problem of stabbing a set of straight line segments with the least cardinality set of disks of fixed radii $r>0$ where the set of segments forms a straight line drawing $G=(V,E)$ of a planar graph without edge crossings. Close geometric problems arise in network security applications. We give strong NP-hardness of the problem for edge sets of Delaunay triangulations, Gabriel graphs and other subgraphs (which are often used in network design) for $r\in [d_{\min},\eta d_{\max}]$ and some constant $\eta$ where $d_{\max}$ and $d_{\min}$ are Euclidean lengths of the longest and shortest graph edges respectively. Fast $O(|E|\log|E|)$-time $O(1)$-approximation algorithm is proposed within the class of straight line drawings of planar graphs for which the inequality $r\geq \eta d_{\max}$ holds uniformly for some constant $\eta>0,$ i.e. when lengths of edges of $G$ are uniformly bounded from above by some linear function of $r.$