Capturing an Evader in Polygonal Environments: A Complete Information Game

TL;DR

This paper proves that three pursuers are always sufficient and sometimes necessary to guarantee the capture of an evader in a polygonal environment with full surveillance, regardless of the environment's complexity.

Contribution

It establishes a complete information pursuit-evasion model in polygonal environments, showing a fixed number of pursuers needed independent of environment complexity.

Findings

Three pursuers suffice for guaranteed capture.

The number of pursuers needed is sometimes necessary, regardless of environment complexity.

The result contrasts with incomplete information scenarios requiring more pursuers.

Abstract

Suppose an unpredictable evader is free to move around in a polygonal environment of arbitrary complexity that is under full camera surveillance. How many pursuers, each with the same maximum speed as the evader, are necessary and sufficient to guarantee a successful capture of the evader? The pursuers always know the evader's current position through the camera network, but need to physically reach the evader to capture it. We allow the evader the knowledge of the current positions of all the pursuers as well---this accords with the standard worst-case analysis model, but also models a practical situation where the evader has "hacked" into the surveillance system. Our main result is to prove that three pursuers are always sufficient and sometimes necessary to capture the evader. The bound is independent of the number of vertices or holes in the polygonal environment. The result should be contrasted with the incomplete information pursuit-evasion where at least {\Omega}(\surd h + log n) pursuers are required just for detecting the evader in an environment with n vertices and h holes.