Art gallery problem
The art gallery problem or museum problem is a well-studied visibility problem in computational geometry. It originates from a real-world problem of guarding an art gallery with the minimum number of guards who together can observe the whole gallery. In the geometric version of the problem, the layout of the art gallery is represented by a simple polygon and each guard is represented by a point in the polygon. A set formula_1 of points is said to guard a polygon if, for every point formula_2 in the polygon, there is some formula_3 such that the  line segment between formula_2 and formula_5 does not leave the polygon. Two dimensions. There are numerous variations of the original problem that are also referred to as the art gallery problem. In some versions guards are restricted to the perimeter, or even to the vertices of the polygon. Some versions require only the perimeter or a subset of the perimeter to be guarded. Solving the version in which guards must be placed on vertices and only vertices need to be guarded is equivalent to solving the dominating set problem on the visibility graph of the polygon. Chvátal's art gallery  theorem. Chvátal's art gallery theorem, named after Václav Chvátal, gives an upper bound on the minimal number of guards. It states that formula_6 guards are always sufficient and sometimes necessary to guard a simple polygon with formula_7 vertices. The question about how many vertices/watchmen/guards were needed was posed to Chvátal by Victor Klee in 1973. Chvátal proved it shortly thereafter. Chvátal's proof was later simplified by Steve Fisk, via a 3-coloring argument. Fisk's short proof. Steve Fisk's proof is so short and elegant that it was chosen for inclusion in "Proofs from THE  BOOK". The proof goes as follows: First, the polygon is triangulated (without adding extra vertices). The vertices of the resulting triangulation graph may be 3-colored. Clearly, under a 3-coloring, every triangle must have all three colors. The vertices with any one color form a valid guard set, because every triangle of the polygon is guarded by its vertex with that color. Since the three colors partition the "n" vertices of the polygon, the color with the fewest vertices defines a valid guard set with at most formula_8 guards. Generalizations. Chvátal's upper bound remains valid if the restriction  to guards at corners is loosened to guards at any point not exterior to the polygon. There are a number of other generalizations and specializations of the original art-gallery theorem. For instance, for orthogonal polygons, those whose edges/walls meet at right angles, only formula_9 guards are needed. There are at least three distinct proofs of this result, none of them simple: by Kahn, Klawe, and Kleitman; by Lubiw; and by Sack and Toussaint. A related problem asks for the number of guards to cover the exterior of an arbitrary polygon (the "Fortress Problem"): formula_10 are sometimes necessary and  always sufficient if guards are placed on the boundary of the polygon, while formula_11 are sometimes necessary and always sufficient if guards are placed anywhere in the exterior of the polygon. In other words, the infinite exterior is more challenging to cover than the finite interior. Computational complexity. In decision problem versions of the art gallery problem, one is given as input both a polygon and a number "k", and must determine whether the polygon can be guarded with "k" or fewer guards. This problem is formula_12-complete, as is the version where the guards are restricted to the edges  of the polygon. Furthermore, most of the other standard variations (such as restricting the guard locations to vertices) are NP-hard. Regarding approximation algorithms for the minimum number of guards, proved the problem to be APX-hard, implying that it is unlikely that any approximation ratio better than some fixed constant can be achieved by a polynomial time approximation algorithm. showed that a logarithmic approximation may be achieved for the minimum number of vertex guards by discretizing the input polygon into convex subregions and then reducing the problem to a set cover problem. As showed, the set system derived from  an art gallery problem has bounded VC dimension, allowing the application of set cover algorithms based on ε-nets whose approximation ratio is the logarithm of the optimal number of guards rather than of the number of polygon vertices. For unrestricted guards, the infinite number of potential guard positions makes the problem even more difficult. However by restricting the guards to lie on a fine grid, a more complicated logarithmic approximation algorithm can be derived under some mild extra assumptions, as shown by . However, efficient algorithms are known for finding a set of at most formula_6 vertex guards, matching Chvátal 's upper bound. For simple polygons that do not contain holes, the existence of a constant factor approximation algorithm for vertex and edge guards was conjectured by Ghosh. Ghosh's conjecture was initially shown to be true for vertex guards in two special sub-classes of simple polygons, viz. monotone polygons and polygons weakly visible from an edge. presented an approximation algorithm that computes in polynomial time a vertex guard set for a monotone polygon such that the size of the guard set is at most 30 times the optimal number of vertex guards. presented an approximation algorithm that computes  in O(n) time a vertex guard set for a simple polygon that is weakly visible from an edge such that the size of the guard set is at most 6 times the optimal number of vertex guards. Subsequently, claimed to have settled the conjecture completely by presenting constant factor approximation algorithms for guarding general simple polygons using vertex guards and edge guards. For vertex guarding the subclass of simple polygons that are weakly visible from an edge, a polynomial-time approximation scheme was proposed by . An exact algorithm was proposed by for vertex guards. The authors conducted extensive computational experiments  with several classes of polygons showing that optimal solutions can be found in relatively small computation times even for instances associated to thousands of vertices. The input data and the optimal solutions for these instances are available for download. Three dimensions. If a museum is represented in three dimensions as a polyhedron, then putting a guard at each vertex will not ensure that all of the museum is under observation. Although all of the surface of the polyhedron would be surveyed, for some polyhedra there are points in the interior which might not be under surveillance. Possible Usage Scenarios of  the Art Gallery Problem The art gallery problem can be used to position multiple cameras when the objective is to cover a large area in view of cameras, using the minimum number of cameras.