New Hardness Results for Guarding Orthogonal Polygons with Sliding Cameras

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 inside 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) 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 asked by Katz and Morgenstern [8]. We next show that the minimum-cardinality sliding cameras problem is NP-hard when P is allowed to have holes, which partially answers another question asked by Katz and Morgenstern [8].