Art Gallery Localization

Wireless Localization with Vertex Guards is NP-hard

We consider a special class of art gallery problems inspired by wireless localization. Given a simple polygon P , place and orient guards each of which broadcasts a unique key within a fixed angular range. In contrast to the classical art gallery setting, broadcasts are not blocked by the boundary of P . At any point in the plane one must be able to tell whether or not one is located inside P o...

Optimal Art Gallery Localization is NP-hard

Art Gallery Localization (AGL) is the problem of placing a set T of broadcast towers in a simple polygon P in order for a point to locate itself in the interior. For any point p ∈ P : for each tower t ∈ T ∩ V (p) (where V (p) denotes the visibility polygon of p) the point p receives the coordinates of t and the Euclidean distance between t and p. From this information p can determine its coordi...

An Efficient Exact Algorithm for the Natural Wireless Localization Problem

Considered a variation of the art gallery problem, the wireless localization problem deals with the placement of the smallest number of broadcasting antennas required to satisfy some property within a given polygon. The case dealt with here consists of antennas that propagate a unique key within a certain antenna-specific angle of broadcast, so that the set of keys received at any given point i...

Art Gallery and Illumination Problems

How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls? This wonderfully na¨ıve question of combinatorial geometry has, since its formulation, stimulated an increasing number of of papers and surveys. In 1987, J. O'Rourke published his book Art Gallery Theorems and Algorithms which has further fueled this area of resear...

The Inapproximability of Illuminating Polygons by α-Floodlights

We consider variants of the art gallery problem where guard visibility is limited to a certain angular aperture α. We show that the problem is NP-hard even when guards can be located in the interior of the polygon. We then proceed to prove that both this problem and its vertex variant, where guard placement is restricted to the vertices of the polygon, are APX-hard. We observe that earlier cons...

Two-Guard Art Gallery Problem