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. 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 in the interior of every edge. We give an algorithm to compute a minimum witness set for P in O(n 2 log n) time, if such a set exists, or to report the non-existence within the same time bounds.