The firefighter problem with more than one firefighter on trees

In this paper we study the complexity of generalized versions of the firefighter problem on trees, and answer several open questions of Finbow and MacGillivray (2009) [8]. More specifically, we consider the version denoted by Max(S,b)(S,b)-Fire where b≥2b≥2 firefighters are allowed at each time step and the objective is to maximize the number of saved vertices that belong to SS. We also study the related decision problem (S,b)(S,b)-Fire that asks whether all the vertices in SS can be saved using b≥2b≥2 firefighters at each time step.We show that (S,b)(S,b)-Fire is NP-complete for trees of maximum degree b+2b+2 even when SS is the set of leaves. Using this last result, we prove the NP-hardness of Max(S,b)(S,b)-Fire for trees of maximum degree b+3b+3 even when SS is the set of all vertices. On the positive side, we give a polynomial-time algorithm for solving (S,b)(S,b)-Fire and Max(S,b)(S,b)-Fire on trees of maximum degree b+2b+2 when the fire breaks out at a vertex of degree at most b+1b+1. Moreover, we present a polynomial-time algorithm for the Max(S,b)(S,b)-Fire problem (and the corresponding weighted version) for a subclass of trees, namely kk-caterpillars. Finally, we observe that the minimization version of Max(S,b)(S,b)-Fire is not n1−εn1−ε-approximable on trees for any ϵ∈(0,1)ϵ∈(0,1) and b≥1b≥1 if P≠NPP≠NP.