Combinatorics and complexity of guarding polygons with edge and point 2-transmitters

We consider a generalization of the classical Art Gallery Problem, where instead of a light source, the guards, called k-transmitters, model a wireless device with a signal that can pass through at most k walls. We show it is NP-hard to compute a minimum cover of point 2transmitters, point k-transmitters, and edge 2-transmitters in a simple polygon. The point 2transmitter result extends to orthogonal polygons. In addition, we give necessity and sufficiency results for the number of edge 2-transmitters in general, monotone, orthogonal monotone, and orthogonal polygons. ∗Abstracts of part of this work appeared in the informal workshops FWCG [7] and EuroCG [8] †Supported by a Clare Boothe Luce Graduate Fellowship and NSF DGE-1148903. ‡Supported by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11). ar X iv :1 50 3. 05 68 1v 1 [ cs .C G ] 1 9 M ar 2 01 5