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

Abstract

A sliding k-transmitter in an orthogonal polygon P is a mobile guard that travels back and forth along an orthogonal line segment s inside P . It can see a point p ∈ P if the perpendicular from p onto s intersects the boundary of P at most k times. We show that guarding an orthogonal polygon P with the minimum number of k-transmitters is NP-hard, for any fixed k > 0, even if P is simple and monotone. Moreover, we give an O(1)-approximation algorithm for this problem.

Guarding monotone art galleries with sliding cameras in linear time

“…Biedl et al [4] studied the MSC problem under k-visibility; that is, the line of sight of a camera can intersect the boundary of the polygon at most k times (note that when k = 0, we have the standard MSC problem studied in this paper).…”

“…The k-visibility has been already studied under the classical art gallery problem [1,2,8]. Biedl et al [4] showed that the MSC problem under k-visibility is NP-hard on simple orthogonal polygons for any k > 0, even if the polygon is monotone. They also gave an O (1)-approximation algorithm for any fixed k > 0.…”

“…Moreover, Eidenbenz [13] proved that the problem is APX-hard on simple polygons. Ghosh [14] gave an O (log n)-approximation algorithm that runs in O (n 4 ) time on simple polygons. King and Kirkpatrick [17] gave an O (log log OPT)-approximation algorithm for the vertex guards (and in fact when the guards can be anywhere on the boundary of the polygon), where OPT is the size of an optimal solution.…”

$k$-Transmitter Watchman Routes

“…Moreover, they gave upper and lower bounds for the number of edge 2-transmitters in general, monotone, orthogonal monotone, and orthogonal polygons; and improved the bound from [14] for simple n-gons to n/5 2-transmitters. For the AGP with k-transmitters, no approximation algorithms have been obtained so far, but Biedl et al [17] recently presented a first constant-factor approximation result for so-called sliding k-transmitters (traveling along an axis-parallel line segment s in the polygon, covering all points p of the polygon for which the perpendicular from p onto s intersects at most k edges of the polygon).…”

“…Of course, k-transmitters do not have to be stationary (or restricted to travel along a special structure as in [17]): we might have to find a shortest tour such that a mobile k-transmitter traveling along this route can establish a connection with all (or a discrete subset of the) points of an environment, the WRP with a k-transmitter. This problem is the focus of this paper and to the best of our knowledge it has not been studied before.…”