Guarding Art Galleries by Guarding Witnesses

Let P be a simple polygon. We define a witness set W to be a set of points such that if any (prospective) guard set G guards W , then it is guaranteed that G guards P . We show that not all polygons admit a finite witness set. If a finite minimal witness set exists, then it cannot contain any witness in the interior of P ; all witnesses must lie on the boundary of P , and there can be at most one witness per edge. We give an algorithm to compute a minimal witness set for P in O(n logn) time, if such a set exists, or to report the non-existence within the same time bounds. We also outline two algorithms that use a witness set for P to test whether a (prospective) guard set sees all points in P .