Minimal Guard Coverage

In Chapter 1 it was shown that Fisk's proof of the art gallery theorem can be converted into an algorithm that covers a polygon of n vertices with [n/3\ guards in O{n logn) time. Although [n/3\ is necessary in some cases, often this is far more guards than are needed to cover a particular polygon. For example, convex polygons only require one guard, but Avis and Toussaint's algorithm would still place [n/3j guards. It is natural, then, to seek a placement of a minimal number of guards that cover a given polygon. We will show, however, that this problem is fundamentally intractable: it is NP-complete. Finding the minimal number of guards to cover a polygon is a specific instance of a general class of problems on which there is now a considerable literature: polygon decomposition problems. Guards determine star polygons, so minimal guard coverage corresponds to finding a minimal star cover of a polygon. Polygon decomposition problems can be classified along four "dimensions": decomposition type, the shape of the pieces, restrictions on the boundaries of the pieces, and the shape of the polygon being decomposed. We will discuss these dimensions briefly before focusing on our particular case.