Multi-guard covers for polygonal regions

We study the problem of finding optimal covers of polygonal regions using multiple mobile guards. By our definition, a point is covered if, at some time, it lies within the convex hull of the guards from which it is visible. The definition captures our desire that guards both “see” and “surround” points that they cover. Guards move along continuous timeparameterized curves within a polygonal region P . An optimal m-guard cover of P is a set of m guard paths of minimum total length that cover all points in P . In this paper, we restrict our attention to the case where P is convex, and m is either two or three. We first address the apparently simpler problem of optimally covering all points on the boundary, ∂P , of P . Although the guard paths are not restricted to ∂P , we prove that in every optimal two-guard boundary cover the guards remain on ∂P . When there are three guards, an optimal boundary cover may require a guard to cross the interior of the polygon. We show, however, that every optimal three-guard boundary cover is simple (i.e., guard paths do not cross one another). We provide complete characterizations of the form of optimal twoand three-guard boundary covers for convex polygons that support polynomial-time algorithms for their construction. Finally, we show that, for convex P , any optimal twoor three-guard cover of ∂P is also a (necessarily optimal) cover of the full polygon P .