Polygon Guarding with Orientation

Date of Submission: 
January 3, 2014
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The art gallery problem is a classical sensor placement problem that asks for the minimum number of guards required to see every point in an environment. The standard formulation does not take into account self-occlusions caused by a person or an object within the environment. Obtaining a good view of an object from all orientations is viewed as an important requirement for surveillance and visual tracking applications. We study the art gallery problem under a constraint, termed triangle-guarding, that ensures that all sides of any convex object are always visible in spite of self-occlusion. Our contributions in this paper are two-fold: we first prove that $\Omega(\sqrt{n})$ guards are always necessary for triangle-guarding the interior of a simple polygon having n vertices. Next, we study the problem of triangle-guarding a set of line segments connecting points on the boundary of the polygon. This is motivated by applications where an object or person of interest can only move along certain paths in the polygon. We present a constant factor approximation algorithm for this problem -- one of the few such results for art gallery problems.