Politician's Firefighting

Firefighting is a combinatorial optimization problem on graphs that models the problem of determining the optimal strategy to contain a fire and save as much (trees, houses, etc.) from the fire as possible. We study a new version of firefighting, which we call Politician’s Firefighting and which exhibits more locality than the classical one-firefighter version. We prove that this locality allows us to develop an O(bn)-time algorithm for this problem on trees, where b is the number of nodes initially on fire. We further prove that Politician’s Firefighting is NP-hard on planar graphs of degree at least 5, and we present an O(m+ k4)-time algorithm for this problem on general graphs, where k is the number of nodes that burn using the optimal strategy, thereby proving that it is fixed-parameter tractable. We present experimental results that show that our algorithm’s search tree size is in practice much smaller than the worst-case bound of 4.