Capture bounds for visibility-based pursuit evasion

We investigate the following problem in the visibility-based discrete-time model of pursuit evasion in the plane: how many pursuers are needed to capture an evader in a polygonal environment with obstacles under the minimalist assumption that pursuers and the evader have the same maximum speed? When the environment is a simply-connected (hole-free) polygon of n vertices, we show that Θ(n1/2) pursuers are both necessary and sufficient in the worst-case. When the environment is a polygon with holes, we prove a lower bound of Ω(n2/3) and an upper bound of O(n5/6) for the number of pursuers that are needed in the worst-case, where n is the total number of vertices including the hole boundaries. More precisely, if the polygon contains h holes, our upper bound is O(n1/2h1/4), for h≤n2/3, and O(n1/3h1/2) otherwise. We then show that with additional assumptions these bounds can be drastically improved. Namely, if the players' movement speed is small compared to the “feature size” of the environment, we give a deterministic algorithm with a worst-case upper bound of O(log⁡n) pursuers for simply-connected n-gons and O(h+log⁡n) for multiply-connected polygons with h holes. Further, if the pursuers are allowed to randomize their strategy, regardless of the players' movement speed, we show that O(1) pursuers can capture the evader in a simply connected n-gon and O(h) when there are h holes with high probability.