A Complexity result for the pursuit-evasion game of maintaining visibility of a moving evader

In this paper we consider the problem of maintaining visibility of a moving evader by a mobile robot, the pursuer, in an environment with obstacles. We simultaneously consider bounded speed for both players and a variable distance separating them. Unlike our previous efforts [R. Murrieta-Cid et al., 2007], we give special attention to the combinatorial problem that arises when searching for a solution through visiting several locations. We approach evader tracking by decomposing the environment into convex regions. We define two graphs: one is called the mutual visibility graph (MVG) and the other the accessibility graph (AG). The MVG provides a sufficient condition to maintain visibility of the evader while the AG defines possible regions to which either the pursuer or the evader may go to. The problem is framed as a non cooperative game. We establish the existence of a solution, based on a k- Min approach, for the following givens: the environment, the initial state of the evader and the pursuer, including their maximal speeds. We show that the problem of finding a solution to this game is NP-complete.