Polygon Guarding With Orientation

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 good views of an object from all orientations despite self-occlusions is an important requirement for surveillance and visual inspection applications. In this talk, we will see a version of the art gallery problem under a constraint, termed △-guarding, that ensures that all sides of any convex object are always visible in spite of self-occlusion.

I will present 3 main results in this talk. First, I will prove that Omega(√n) guards are always necessary for △-guarding the interior of a simple polygon having n vertices. Second, I will present a O(log c_{opt}) factor approximation algorithm for △-guarding polygons with or without holes, when the guards are restricted to vertices of the polygon, where c_{opt} is the optimal number of guards. Third, I will introduce the problem of △-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. If time permits, I will outline a constant factor approximation algorithm for this problem – one of the few such results for art gallery problems.