Before revision:  We present a polynomial time algorithm for both problems, and show that the cardinality of a minimum guard set and the cardinality of a maximum witness set coincide.Thus, uni-monotone polygons are perfect; this result also extends to monotone mountains .
After revision:  We present a polynomial time algorithm for both problems, and show that the cardinality of a minimum guard set and the cardinality of a maximum number of ``witnesses'', i.e., terrain points, no two of which can be seen by a single guard. We show that our results also apply to the Art Gallery problem in ``monotone mountains'', i.e., x-monotone polygons with a single edge as one of the boundary chains. This means that any monotone mountain is ``perfect'' (its guarding number is the same as its witness number); we thus establish the first non-trivial class of perfect polygons .