Escaping a Polygon

TL;DR

This paper models a pursuit-evasion game within polygons, establishing conditions for escape based on speed ratios, providing exact and approximate solutions for simple polygons, and proving computational hardness for complex domains.

Contribution

It formalizes a pursuit-evasion game avoiding pathological cases, derives exact critical speed ratios for specific polygons, and develops efficient approximation algorithms, while also proving hardness results for more complex scenarios.

Findings

Exact critical speed ratios for equilateral triangle and square.

Polynomial-time algorithm for approximating the critical speed ratio.

NP-hardness and PSPACE-hardness results for complex and multi-player scenarios.

Abstract

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.