The Containment Game in the plane: between the Firefighter Problem and Conway's Angel Problem

TL;DR

This paper introduces a new containment game that generalizes the Firefighter Problem and Conway's Angel Problem, analyzing the minimal growth rate of spreading to prevent perpetual spread.

Contribution

It defines the containment game framework, relates it to known problems, and establishes bounds on the spreading function for winning strategies.

Findings

Established a sub-linear upper bound g(t)=O(t^{6/7}) for containment.

Proved a lower bound g(t)=Ω(t^{1/2}) for the spreading function.

Provided explicit strategies demonstrating these bounds.

Abstract

The containment game is a full information game for two players, initialised with a set of occupied vertices in an infinite connected graph $G$. On the $t$-th turn, the first player, called Spreader, extends the occupied set to $g(t)$ adjacent vertices, and then the second player, called Container, removes $q$ unoccupied vertices from the graph. If the spreading process continues perpetually -- Spreader wins, and otherwise -- Container wins. For $g=\infty$ this game reduces to a solitaire game for Container, known as the Firefighter Problem. On $\mathbb{Z}^2$, for $q=1/k$ and $g\equiv 1$ it is equivalent to Conway's Angel Problem. We introduce the game, and writing $q(G,g)$ for the set of $q$ values for which Container wins against a given $g(t)$, we study the minimal asymptotics of $g(t)$ such that $q(G,g)=q(G,\infty)$, i.e. for which defeating Spreader is as hard as winning the Firefighter Problem solitaire. We show, by providing explicit winning strategies, a sub-linear upper bound $g(t)=O(t^{6/7})$ and a lower bound of $g(t)=\Omega(t^{1/2})$.