Stabbing Line Segments with Disks: Complexity and Approximation Algorithms

Computational complexity is studied for the problem of stabbing set of straight line segments with the smallest cardinality set of disks of fixed radii r > 0 where the set of segments forms straight line drawing of planar graph. This problem along with its relatives [3] arise in physical network security analysis for telecommunication, wireless and road networks represented by geometric graphs based on euclidean distances between their vertices (proximity graphs). Among those proximity graphs are Delaunay triangulations, its subgraphs and half-θ6 graphs [7] which admit efficient geometric routing. Being of particular interest computational complexity of this problem did not receive much attention in the literature. In this paper we claim strong NP-hardness of the problem over the classes of 4-connected (i.e. Hamiltonian) plane triangulations of bounded vertex degree as well as of 4connected plane half-θ6 graphs for small r. It remains strongly NP-hard over the classes of Delaunay triangulations and some of their connected subgraphs (Gabriel and relative neighbourhood graphs) for values of r of the same scale as minimum graph edge length whereas for large r the problem becomes polynomially solvable over connected plane graphs.