A Leapfrog Strategy for Pursuit-Evasion in a Polygonal Environment

Abstract

We study pursuit-evasion in a polygonal environment with polygonal obstacles. In this turn based game, an evader [Formula: see text] is chased by pursuers [Formula: see text]. The players have full information about the environment and the location of the other players. The pursuers are allowed to coordinate their actions. On the pursuer turn, each [Formula: see text] can move to any point at distance at most 1 from his current location. On the evader turn, he moves similarly. The pursuers win if some pursuer becomes co-located with the evader in finite time. The evader wins if he can evade capture forever. It is known that one pursuer can capture the evader in any simply-connected polygonal environment, and that three pursuers are always sufficient in any polygonal environment [Formula: see text] (possibly with polygonal obstacles). We contribute two new results to this field. First, we fully characterize when an environment with a single obstacle is one-pursuerwin or two-pursuer-win. Second, we give sufficient (but not necessary) conditions for an environment to have a winning strategy for two pursuers. Such environments can be swept by a leapfrog strategy in which the two cops alternately guard/increase the currently controlled area. The running time of this algorithm is [Formula: see text] where [Formula: see text] is the number of vertices, [Formula: see text] is the number of obstacles and [Formula: see text] is the diameter of the polygonal environment [Formula: see text]. More concretely, for an environment with [Formula: see text] vertices, we describe an [Formula: see text] algorithm that (1) determines whether the obstacles are well-separated, and if so, (2) constructs the required partition for a leapfrog strategy.