Guarding orthogonal art galleries with sliding cameras

Abstract Let P be an orthogonal polygon with n vertices. A sliding camera travels back and forth along an orthogonal line segment s ⊆ P corresponding to its trajectory. The camera sees a point p ∈ P if there is a point q ∈ s such that p q ‾ is a line segment normal to s that is completely contained in P. In the Minimum-Cardinality Sliding Cameras (MCSC) 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 (MLSC) 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 answer questions posed by Katz and Morgenstern (2011) by presenting the following results: (i) the MLSC problem is polynomially tractable even for orthogonal polygons with holes, (ii) the MCSC problem is NP -complete when P is allowed to have holes, and (iii) an O ( n 3 log ⁡ n ) -time 2-approximation algorithm for the MCSC problem on [NE]-star-shaped orthogonal polygons with n vertices (similarly, [NW]-, [SE]-, or [SW]-star-shaped orthogonal polygons).

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