The Searchlight Scheduling Problem

The problem of searching for a mobile robber in a simple polygon by a number of searchlights is considered. A searchlight is a stationary point which emits a single ray that cannot penetrate the boundary of the polygon. The direction of the ray can be changed continuously, and a point is detected by a searchlight at a given time if and only if it is on the ray. A robber is a point that can move continuously with unbounded speed. First, it is shown that the problem of obtaining a search schedule for an instance having at least one searchlight on the polygon boundary can be reduced to that for instances having no searchlight on the polygon boundary. The reduction is achieved by a recursive search strategy called the one-way sweep strategy. Then various sufficient conditions for the existence of a search schedule are presented by using the concept of a searchlight visibility graph. Finally, a simple necessary and sufficient condition for the existence of a search schedule for instances having exactly two searchlights in the interior is presented. Key words, geometry, searchlight, visibility AMS(MOS) subject classification. 68E99