Fire containment in grids of dimension three and higher

We consider a deterministic discrete-time model of fire spread introduced by Hartnell [1995] and the problem of minimizing the number of burnt vertices when deploying a limited number of firefighters per timestep. While only two firefighters per timestep are needed in the two dimensional lattice to contain any outbreak, we prove a conjecture of Wang and Moeller [2002] that 2d − 1 firefighters per timestep are needed to contain a fire outbreak starting at a single vertex in the d-dimensional square lattice for d ≥ 3; we also prove that in the d-dimensional lattice, d ≥ 3, for each positive integer f there is some outbreak of fire such that f firefighters per timestep are insufficient to contain the outbreak. We prove another conjecture of Wang and Moeller that the proportion of elements in the three-dimensional grid Pn ×Pn ×Pn which can be saved with one firefighter per time step when an outbreak starts at one vertex goes to 0 as n gets large. Finally, we use integer programming to prove results about the minimum number of timesteps needed and minimum number of burnt vertices when containing a fire outbreak in the two dimensional square lattice with two firefighters per timestep. Mathematics Subject Classification: 05C75, 90C35