Parameterized complexity of firefighting

The Firefighter problem is to place firefighters on the vertices of a graph to prevent a fire with known starting point from lighting up the entire graph. In each time step, a firefighter may be placed on an unburned vertex, permanently protecting it, and the fire spreads to all neighboring unprotected vertices of burning vertices. The goal is to let as few vertices burn as possible. This problem is known to be NP-complete, even when restricted to trees of maximum degree three. Initial study showed the Firefighter problem to be fixed-parameter tractable on trees in various parameterizations. In this paper, we consider a generalization of this problem, where at each time step b ≥ 1 firefighters can be deployed. Our results answer several open questions raised by Cai et al. [8]. We show that this problem is W[1]-hard when parameterized by the number of saved vertices, protected vertices, and burned vertices. We also investigate several combined parameterizations for which the problem is fixed-parameter tractable. Some of our algorithms improve on previously known algorithms. We also establish lower bounds to polynomial kernelization.