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.Comment: 8 page
