Guarding Orthogonal Art Galleries Using Sliding Cameras: Algorithmic and Hardness Results

Abstract

Let P be an orthogonal polygon. Consider a sliding camera that travels back and forth along an orthogonal line segment s ⊆ P as its trajectory. The camera can see a point p ∈ P if there exists a point q ∈ s such that pq is a line segment normal to s that is completely contained in P . In the minimum-cardinality sliding cameras problem, the objective is to find a set S of sliding cameras of minimum cardinality to guard P (i.e., every point in P can be seen by some sliding camera in S) while in the minimum-length sliding cameras problem the goal is to find such a set S so as to minimize the total length of trajectories along which the cameras in S travel. In this paper, we first settle the complexity of the minimum-length sliding cameras problem by showing that it is polynomial tractable even for orthogonal polygons with holes, answering a question posed by Katz and Morgenstern [9]. Next we show that the minimum-cardinality sliding cameras problem is NP-hard when P is allowed to have holes, which partially answers another question posed by Katz and Morgenstern [9].

On Guarding Orthogonal Polygons with Sliding Cameras

“…These problems were introduced by Katz and Morgenstern [19], who proved that MHSC can be solved in polynomial time in the special case where the polygon is simple (has no holes). It was shown later that MSC is NP-hard in polygons with holes [12]; NP-hardness in simple polygons is open. Durocher et al [11] claimed a (3.5)-approximation algorithm for MSC problem on simple orthogonal polygons, but this was later discovered by the authors to be incorrect (private communication).…”

Guarding Monotone Art Galleries with Sliding Cameras in Linear Time

“…Later Durocher et al [2] gave an O(n 2.5 )-time (3.5)-approximation algorithm for the MSC problem on simple orthogonal polygons. Durocher and Mehrabi [3] showed that the MSC problem is np-hard when the polygon P is allowed to have holes. Durocher and Mehrabi also considered a variant of the problem, called the MLSC problem, in which the objective is to minimize the sum of the lengths of line segments along which cameras travel, and proved that the MLSC problem is polynomial-time solvable even on orthogonal polygons with holes.…”

Mobile versus Point Guards

“…Finding a minimum cardinality mixed vertical and horizontal mobile r-guard system (also known as the Minimum cardinality Sliding Cameras or MSC problem) has been shown by Durocher and Mehrabi [6] to be NP-hard for orthogonal polygons with holes. For orthogonal polygons without holes, the problem translates to the Dominating Set problem in the pixelation graph.…”