The Firefighter Problem: A Structural Analysis

  • Janka Chlebíková
  • Morgan ChopinEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8894)


We consider the complexity of the firefighter problem where a budget of \({b \ge 1}\) firefighters are available at each time step. This problem is proved to be NP-complete even on trees of degree at most three and \({b = 1}\) [10] and on trees of bounded degree \(b+3\) for any fixed \(b \ge 2\) [3]. In this paper, we provide further insight into the complexity landscape of the problem by showing a complexity dichotomy result with respect to the parameters pathwidth and maximum degree of the input graph. More precisely, we first prove that the problem is NP-complete even on trees of pathwidth at most three for any \(b \ge 1\). Then we show that the problem turns out to be fixed parameter-tractable with respect to the combined parameter “pathwidth” and “maximum degree” of the input graph. Finally, we show that the problem remains NP-complete on very dense graphs, namely co-bipartite graphs, but is fixed-parameter tractable with respect to the parameter “cluster vertex deletion”.


Maximum Degree Vertex Cover Input Graph Dense Graph Unit Disk Graph 
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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.University of PortsmouthSchool of ComputingPortsmouthUK
  2. 2.Institut für Optimierung und Operations ResearchUniversität UlmUlmGermany

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