逃离多边形
Escaping a Polygon
AI总结：
本文研究区域内外双方最优追逃的临界速度比，建立唯一胜者模型，精确求解三角形与正方形，并给出简单多边形的近似算法及三维、多玩家情形的复杂性下界。
摘要：
假设一名逃跑玩家（“人类”）在一个区域的内部以最大速度 $1$ 连续移动，而一名追捕玩家（“僵尸”）在该区域外部以最大速度 $r$ 连续移动。在双方均采取最优策略的前提下，当 $r$ 取何值时，第一名玩家能够逃离该区域，也就是在与追捕玩家保持正距离的情况下到达边界？我们形式化了一个用于这种无限小交替的双人博弈模型，并证明它在任何局部可求长区域中都有唯一的胜者。因此，我们的模型避免了此前在某些度量空间中的 Lion and Man（狮子与人）等追逃博弈里发现的病态行为（即两名玩家都可能拥有“获胜策略”）。对于若干特定区域，包括等边三角形和正方形，我们给出了临界速度比的精确结果：高于该比值时追捕玩家可以获胜，低于该比值时逃跑玩家可以获胜（而恰好在该比值时追捕玩家可以获胜）。对于简单多边形，我们给出了一个简单公式和多项式时间算法，保证可得到临界速度比的 10.89898 近似，并给出了一个伪多项式时间近似方案，可任意精确地逼近临界速度比。在负面结果方面，我们证明了三维多面体区域中该问题的 NP 困难性；对于推广到多名逃跑玩家和追捕玩家的情形，我们还证明了更强的结果（PSPACE 困难性，甚至近似也是 NP 困难的）。
英文摘要：
Suppose an escaping player ("human") moves continuously at maximum speed $1$ in the interior of a region, while a pursuing player ("zombie") 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 and prove that it has a unique winner in any locally rectifiable region. Our model thus avoids 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 specific 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 approximating the critical speed ratio arbitrarily closely. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
