On Guarding Orthogonal Polygons with Sliding Cameras

Abstract

A sliding camera inside an orthogonal polygon P is a point guard that travels back and forth along an orthogonal line segment γ in P . The sliding camera g can see a point p in P if the perpendicular from p onto γ is inside P . In this paper, we give the first constant-factor approximation algorithm for the problem of guarding P with the minimum number of sliding cameras. Next, we show that the sliding guards problem is linear-time solvable if the (suitably defined) dual graph of the polygon has bounded treewidth. Finally, we study art gallery theorems for sliding cameras, thus, give upper and lower bounds in terms of the number of guards needed relative to the number of vertices n.

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

“…As a second result, we show that the O(1)-approximation algorithm that we recently developed for sliding cameras [4] works similarly for sliding k-transmitters. Hence we have an O(1)approximation for finding the minimum set of sliding k-transmitters, in any (not necessarily simple) orthogonal polygon.…”

“…Sliding cameras were introduced by Katz and Morgenstern [9]. Finding the minimum set of sliding cameras is NP-hard in polygons with holes [8], even if only horizontal sliding cameras are allowed [4]. The optimum set of sliding cameras can be found in polynomial time for monotone polygons [7].…”

Mobile versus Point Guards

“…Also, for an orthogonal polygon of n vertices, a covering set of mobile guards of cardinality at most ⌊(3n + 4)/16⌋ (which is the extremal bound shown by Aggarwal [1]) can be found in linear time [12]. In case holes are allowed, [2] give a polynomial time constant factor approximation algorithm.…”

“…In orthogonal polygons with holes, the problem is NP-hard as shown by Biedl, Chan, Lee, Mehrabi, Montecchiani, and Vosoughpour [2]. In their paper, a polynomial time constant factor approximation algorithm for the MHSC problem is described, too.…”

On Guarding Orthogonal Polygons with Bounded Treewidth

“…Therefore, one can argue that the auxiliary graph has bounded treewidth if the polygon does, and so restricted-distance-2 dominating set can be solved. A similar (and even simpler) approach works for sliding cameras [3]; here the "intermediaries" are maximal orthogonal line segments.…”