Catching a fast robber on the grid

TL;DR

This paper investigates the cops and robbers game on a grid, demonstrating that a large constant robber speed significantly increases the number of cops needed to catch the robber, with the required cops growing exponentially.

Contribution

It proves that for high-speed robbers, the number of cops needed grows exponentially with the grid size, extending understanding of pursuit-evasion dynamics.

Findings

Number of cops needed grows exponentially with grid size for high-speed robbers

Two cops suffice when robber speed is one, but more are needed as speed increases

Robber speed significantly impacts pursuit complexity on grids

Abstract

We study the problem of cops and robbers on the grid where the robber is allowed to move faster than the cops. It is well known that two cops are necessary and sufficient to catch the robber on any finite grid when the robber has unit speed. Here, we prove that when the speed of the robber is a sufficiently large constant, the number of cops needed to catch the robber on an $n \times n$ grid is $\exp(\Omega(\log n / \log \log n))$.