Escaping a Polygon

TL;DR

This work formalizes the pursuit--escape problem, a no-capture variant where an interior escaper and an exterior pursuer move at speeds 1 and r, respectively, and the escaper aims to reach a boundary exit at a positive distance from the pursuer. It provides a rigorous model ensuring a unique winner for locally rectifiable domains, introduces delta-oblivious strategies to guarantee well-defined play, and develops a discretization framework that links continuous dynamics to a pseudopolynomial-time approximation scheme. The paper delivers exact critical speed ratios for key Jordan shapes (e.g., wedges, halfplanes, disks, triangles, and squares) and a constant-factor O(1)-approximation algorithm for simple polygons, alongside a PTAS and NP-hardness/PSPACE-hardness results in 3D and multi-agent variants. Collectively, these results advance both the theoretical understanding of pursuit--escape dynamics and practical computation of escape thresholds, with implications for robotics, surveillance, and security in polygonal and polyhedral environments.

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.

Figures (23)

• Figure 1: Simple (suboptimal) strategies for the escaper in two domains: start at $p$, and run at full speed along the dotted shortest path to $q$. The speed ratio $r$ must be at least $d_z(p,q) \over d_h(p,q)$ for the pursuer to thwart this strategy, and thus the critical speed ratio is at least this large; see Theorem \ref{['lower bound']}.
• Figure 2: A polygon and its medial axis.
• Figure 3: The scenario when the escaper leaves the fridge (blue) of a medial-axis region $R$ (red), at a point $h$ now closest to facet $f_q$.
• Figure 4: Two "hourglasses", one inside and the other outside $P$, representing a region $S_1 \times S_2$ where shortest paths (inside or outside $P$) between $S_1$ and $S_2$ are in the same equivalence class.
• Figure 5: Geometry of winning strategies in a wedge [Left] and halfplane [Right].

Theorems & Definitions (89)

• Corollary 2.0
• Theorem 3.1: lower bound
• proof
• Theorem 3.2: upper bound
• proof
• Theorem 3.3
• proof
• Lemma 3.4
• proof
• Theorem 4.1