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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 (√n) 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(n1/2 h1/4), for h ≤ n2/3, and O(n1/3 h1/2) otherwise. These bounds show that capture with minimal assumptions requires significantly more pursuers than what is possible either for visibility detection where pursuers win if one of them can see the evader [Guibas et al. 1999], or for capture when players' movement speed is small compared to "features" of the environment [Klein and Suri, 2012].