Stabbing line segments with disks: Complexity and approximation algorithms

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∈ [ dmin, ηdmax] and some constant η where dmax and dmin 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≥ ηdmax holds uniformly for some constant η> 0, i.e. when lengths of edges of G are uniformly bounded from above by some linear function of r.