Making Life Easier for Firefighters
Being a firefighter is a tough job, especially when tight city budgets do not allow enough firefighters to be on duty when a fire starts. This is formalized in the Firefighter problem, which aims to save as many vertices of a graph as possible from a fire that starts in a vertex and spreads through the graph. In every time step, a single additional firefighter may be placed on a vertex, and the fire advances to each vertex in its neighborhood that is not protected by a firefighter. The problem is notoriously hard: it is NP-hard even when the input graph is a bipartite graph or a tree of maximum degree 3, it is W-hard when parameterized by the number of saved vertices, and it is NP-hard to approximate within n1 − ε for any ε > 0. We aim to simplify the task of a firefighter by providing algorithms that show him/her how to efficiently fight fires in certain types of networks. We show that Firefighter can be solved in polynomial time on various well-known graph classes, including interval graphs, split graphs, permutation graphs, and Pk-free graphs for fixed k. On the negative side, we show that the problem remains NP-hard on unit disk graphs.
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