The firefighter problem with more than one firefighter on trees

Authors:
Cristina Bazgan;Morgan Chopin;Bernard Ries
Affiliations:
Université Paris-Dauphine, LAMSADE, Place du Marchal de Lattre de Tassigny, 75775 Paris Cedex 16, France and Institut Universitaire de France, France;Université Paris-Dauphine, LAMSADE, Place du Marchal de Lattre de Tassigny, 75775 Paris Cedex 16, France;Université Paris-Dauphine, LAMSADE, Place du Marchal de Lattre de Tassigny, 75775 Paris Cedex 16, France
Venue:
Discrete Applied Mathematics
Year:
2013

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Abstract

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)-Fire where b=2 firefighters are allowed at each time step and the objective is to maximize the number of saved vertices that belong to S. We also study the related decision problem (S,b)-Fire that asks whether all the vertices in S can be saved using b=2 firefighters at each time step. We show that (S,b)-Fire is NP-complete for trees of maximum degree b+2 even when S is the set of leaves. Using this last result, we prove the NP-hardness of Max(S,b)-Fire for trees of maximum degree b+3 even when S is the set of all vertices. On the positive side, we give a polynomial-time algorithm for solving (S,b)-Fire and Max(S,b)-Fire on trees of maximum degree b+2 when the fire breaks out at a vertex of degree at most b+1. Moreover, we present a polynomial-time algorithm for the Max(S,b)-Fire problem (and the corresponding weighted version) for a subclass of trees, namely k-caterpillars. Finally, we observe that the minimization version of Max(S,b)-Fire is not n^1^-^@e-approximable on trees for any @e@?(0,1) and b=1 if PNP.