Catch Me If You Can: Pursuit and Capture in Polygonal Environments with Obstacles

We resolve a several-years old open question in visibility-based pursuit evasion: how many pursuers are needed to capture an evader in an arbitrary polygonal environment with obstacles? The evader is assumed to be adversarial, moves with the same maximum speed as pursuers, and is "sensed" by a pursuer only when it lies in line-of-sight of that pursuer. The players move in discrete time steps, and the capture occurs when a pursuer reaches the position of the evader on its move. Our main result is that O(√h + log n) pursuers can always win the game with a deterministic search strategy in any polygon with n vertices and h obstacles (holes). In order to achieve this bound, however, we argue that the environment must satisfy a minimum feature size property, which essentially requires the minimum distance between any two vertices to be of the same order as the speed of the players. Without the minimum feature size assumption, we show that Ω(√n/ log n) pursuers are needed in the worst-case even for simply-connected (hole-free) polygons of n vertices! This reveals an unexpected subtlety that seems to have been overlooked in previous work claiming that O(log n) pursuers can always win in simply-connected n-gons. Our lower bound also shows that capturing an evader is inherently more difficult than just "seeing" it because O(log n) pursuers are provably sufficient for line-of-sight detection even against an arbitrarily fast evader in simple n-gons.