Watchman routes in the presence of a pair of convex polygons

Given a set of polygonal obstacles in the plane, the shortest watchman route problem asks for a closed route from which each point in the exterior of the polygons is visible to some point along the route. This problem is known to be NP-hard and the development of an eecient approximation algorithm is still open. We present an O(n 2) time algorithm for computing the shortest watchman route for a pair of convex polygons, where n is the total number of edges in the polygons. We also show that the algorithm can be easily extended, without increasing its complexity, to compute the shortest watchman route when the polygons are enclosed by a third convex polygon.