Firefighting on Trees: (1 - 1/e)---Approximation, Fixed Parameter Tractability and a Subexponential Algorithm

  • Authors:

  • Leizhen Cai;Elad Verbin;Lin Yang

  • Affiliations:

  • Department of Computer Science and Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong SAR, China;The Institute For Theoretical Computer Science, Tsinghua University, Beijing, China;Department of Computer Science and Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong SAR, China

  • Venue:
  • ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
  • Year:
  • 2008

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Abstract

The firefighter problem is defined as follows. Initially, a firebreaks out at a vertex r of a graph G. In eachsubsequent time unit, a firefighter chooses a vertex not yet onfire and protects it, and the fire spreads to all unprotectedneighbors of the vertices on fire. The objective is to choose asequence of vertices for the firefighter to protect so as to savethe maximum number of vertices. The firefighter problem can be usedto model the spread of fire, diseases, computer viruses andsuchlike in a macro-control level.In this paper, we study algorithmic aspects of the firefighterproblem on trees, which is NP-hard even for trees of maximum degree3. We present a (1 - 1/e)-approximation algorithm based onLP relaxation and randomized rounding, and give several FPTalgorithms using a random separation technique of Cai, Chan andChan. Furthermore, we obtain an $2^{O(\sqrt{n}\log n)}$-timesubexponential algorithm.