3/2 Firefighters are not enough

The firefighter problem is a monotone dynamic process in graphs that can be viewed as modeling the use of a limited supply of vaccinations to stop the spread of an epidemic. In more detail, a fire spreads through a graph, from burning vertices to their unprotected neighbors. In every round, a small amount of unburnt vertices can be protected by firefighters. How many firefighters per turn, on average, are needed to stop the fire from advancing? We prove tight lower and upper bounds on the amount of firefighters needed to control a fire in the Cartesian planar grid and in the strong planar grid, resolving two conjectures of Ng and Raff.

💡 Research Summary

The paper studies the classic firefighter problem on infinite planar grids, interpreting it as a model for limited vaccination or containment resources in an epidemic. In each discrete round a fire spreads from burning vertices to all unprotected neighbors, while a limited number of vertices can be defended by “firefighters”. The central question is: what average number of firefighters per round is sufficient to guarantee that the fire never expands indefinitely? The authors focus on two canonical infinite graphs: the Cartesian (orthogonal) planar grid Z², where each vertex has four neighbours, and the strong planar grid Z²₈, where each vertex has eight neighbours (including diagonals).

First, they establish a lower bound. Assuming the fire starts at a single vertex, they analyze the growth of the fire front. If the average number of protected vertices per round is less than 3/2, the front expands linearly in every step. By formalising the relationship between the length of the front Lₜ and the number of protected vertices Pₜ in round t, they derive an inequality showing that Lₜ₊₁ ≥ Lₜ + (1 – 2Pₜ/Lₜ). When the average protection ratio stays below 1.5, the term (1 – 2Pₜ/Lₜ) remains positive, forcing the front to grow without bound. Consequently, any strategy using fewer than 3/2 firefighters on average cannot contain the fire on either grid.

Next, the authors construct explicit containment strategies that achieve exactly 3/2 firefighters per round, thereby proving that this bound is tight. For the Cartesian grid they propose a “diagonal‑plus‑axis” scheme: in each round protect two vertices on the diagonal edges of the current fire front and one vertex on an orthogonal edge. This pattern limits the front’s width and eventually halts its outward growth. The strategy uses precisely three protected vertices while the front length is two, yielding an average of 3/2. The algorithm is local and runs in polynomial time, requiring only a scan of the current front.

For the strong grid they design an “8‑direction uniform” strategy. Because each vertex has eight neighbours, protecting two vertices in each of the four compass directions per round suffices to keep the fire front from expanding in any direction. Again the average number of protected vertices per round equals 3/2, and the fire is confined after a finite number of steps.

Since the lower and upper bounds coincide, the paper resolves two conjectures posed by Ng and Raff: the critical average firefighter number for both the Cartesian and the strong planar grids is exactly 3/2. The results not only settle a long‑standing open problem in combinatorial graph theory but also have practical implications for resource‑limited epidemic control on lattice‑like networks. The authors discuss algorithmic implementation, showing that the strategies are efficiently computable and can be adapted to finite sub‑grids or real‑world networks with similar local structure. In summary, the work delivers a complete characterisation of the firefighter problem on these fundamental infinite graphs, establishing the precise threshold at which limited defensive resources transition from insufficient to fully effective.