Multi-layered planar firefighting

TL;DR

This paper analyzes a fire containment problem on layered grid graphs, determining the minimum protection rate needed to prevent infinite fire spread, and shows that a layered approach is asymptotically optimal.

Contribution

It establishes the critical protection rates for fire containment on layered grid graphs and demonstrates the asymptotic optimality of a multi-layered strategy.

Findings

Critical protection rate for nearest neighbor adjacency: 1.5h

Critical protection rate for strong adjacency: 3h

Layered strategy is asymptotically optimal

Abstract

Consider a model of fire spreading through a graph; initially some vertices are burning, and at every given time-step fire spreads from burning vertices to their neighbours. The firefighter problem is a solitaire game in which a player is allowed, at every time-step, to protect some non-burning vertices (by effectively deleting them) in order to contain the fire growth. How many vertices per turn, on average, must be protected in order to stop the fire from spreading infinitely? Here we consider the problem on $\mathbb{Z}^2\times [h]$ for both nearest neighbour adjacency and strong adjacency. We determine the critical protection rates for these graphs to be $1.5h$ and $3h$, respectively. This establishes the fact that using an optimal two-dimensional strategy for all layers in parallel is asymptotically optimal.