Problems of posting sentries: Variations on the art gallery theorem
We examine the following problem and its variations: “Given a collection of mutually disjoint polygons (the objects) properly contained in a polygon (the enclosure), what is the minimum number of stationary guards that need to be posted within the enclosure but outside the objects so that every edge of each object is seen by some guard”.
Algorithms to compute minimal postings for a convex object in a convex enclosure and a convex object in a nonconvex enclosure are presented. We prove that the problems of computing minimal postings for both a collection of convex objects in a convex enclosure, and a single nonconvex object in a convex enclosure, are NP-Hard.
The variation of posting a given number of guards for a convex object in a convex enclosure so as to minimize a measure of proximity of guards to points of the object is considered. Results for two different measures of proximity are presented.
Finally we show that the 3-dimensional problem of computing a posting with a minimum number of guards for a convex polyhedral object in a convex polyhedral enclosure is NP-Hard. Upperbounds on the number of guards required to cover every point on the surface of the object are derived.
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