Guarding Orthogonal Art Galleries with Sliding k-Transmitters: Hardness and Approximation

A sliding k -transmitter inside an orthogonal polygon P , for a fixed k ≥ 0 , is a point guard that travels along an axis-parallel line segment s in P . The sliding k -transmitter can see a point p ∈ P if the perpendicular from p onto s intersects the boundary of P in at most k points. In the Minimum Sliding k -Transmitters ( ST k ) problem, the objective is to guard P with the minimum number of sliding k -transmitters. In this paper, we give a constant-factor approximation algorithm for the ST k problem on P for any fixed k ≥ 0 . Moreover, we show that the ST 0 problem is NP -hard on orthogonal polygons with holes even if only horizontal sliding 0-transmitters are allowed. For k > 0 , the problem is NP -hard even in the extremely restricted case where P is simple and monotone. Finally, we study art gallery theorems; i.e., we give upper and lower bounds on the number of sliding transmitters required to guard P relative to the number of vertices of P .