Suppose an "escaping" player moves continuously at maximum speed 1 in the interior of a region, while a "pursuing" player moves continuously at maximum speed $r$ outside the region. For what $r$ can the first player escape the region, that is, reach the boundary a positive distance away from the pursuing player, assuming optimal play by both players? We formalize a model for this infinitesimally alternating 2-player game that we prove has a unique winner in any region with locally rectifiable boundary, avoiding pathological behaviors (where both players can have "winning strategies") previously identified for pursuit-evasion games such as the Lion and Man problem in certain metric spaces. For some regions, including both equilateral triangle and square, we give exact results for the critical speed ratio, above which the pursuing player can win and below which the escaping player can win (and at which the pursuing player can win). For simple polygons, we give a simple formula and polynomial-time algorithm that is guaranteed to give a 10.89898-approximation to the critical speed ratio, and we give a pseudopolynomial-time approximation scheme for arbitrarily approximating the critical speed ratio. On the negative side, we prove NP-hardness of the problem for polyhedral domains in 3D, and prove stronger results (PSPACE-hardness and NP-hardness even to approximate) for generalizations to multiple escaping and pursuing players.
翻译:假设一个“ 围观” 玩家在一个区域的内部以最大速度1 移动, 而一个“ 围观” 玩家在一个区域的内部以最大速度1 不断移动, 而一个“ 围观” 玩家在区域之外以最高速度1 美元 持续移动。 对于第一个玩家能够逃离该区域, 也就是说, 到达距离追逐玩家的积极距离, 假设两个玩家都能最佳地玩游戏? 我们正式确定了这个极小的交替的 2玩家游戏的模式, 这个模式在任何具有本地可校正边界的区域都有一个独特的赢家, 避免先前为追逐 - 躲避游戏而确定的病理行为( 在那里, 两个玩家都可以有“ 组合战略 ” ) 。 对于某些区域, 包括等边三角和方, 我们给出关键速度比率的准确结果, 超过这个关键速度比率, 追逐的玩家可以赢取胜出( 和 追击玩家可以赢赢赢赢的游戏家可以赢赢得 ) 。 对于简单的多边形, 我们给出一个简单的公式和多球时段算,, 它保证给关键速度比率的10.898 和对关键速度比率,, 的递增速度, 甚至的轨道, 我们给出了 快速的轨道的轨道的轨道 直对正弦的轨道 。