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

  • Leizhen Cai
  • Elad Verbin
  • Lin Yang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5369)

Abstract

The firefighter problem is defined as follows. Initially, a fire breaks out at a vertex r of a graph G. In each subsequent time unit, a firefighter chooses a vertex not yet on fire and protects it, and the fire spreads to all unprotected neighbors of the vertices on fire. The objective is to choose a sequence of vertices for the firefighter to protect so as to save the maximum number of vertices. The firefighter problem can be used to model the spread of fire, diseases, computer viruses and suchlike in a macro-control level.

In this paper, we study algorithmic aspects of the firefighter problem on trees, which is NP-hard even for trees of maximum degree 3. We present a (1 − 1/e)-approximation algorithm based on LP relaxation and randomized rounding, and give several FPT algorithms using a random separation technique of Cai, Chan and Chan. Furthermore, we obtain an \(2^{O(\sqrt{n}\log n)}\)-time subexponential algorithm.

Keywords

Approximation Ratio Full Paper Fractional Strategy Computer Virus Satisfying Strategy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [AS92]
    Alon, N., Spencer, J.H.: The Probabilistic Method. Wiley, Chichester (1992)zbMATHGoogle Scholar
  2. [CCC06]
    Cai, L., Chan, S.M., Chan, S.O.: Random separation: A new method for solving fixed-cardinality optimization problems. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 239–250. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. [CW07]
    Cai, L., Wang, W.: The surviving rate of a graph (manuscript, 2007)Google Scholar
  4. [DH07]
    Develin, M., Hartke, S.G.: Fire containment in grids of dimension three and higher. Discrete Appl. Math. 155(17), 2257–2268 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [FKMR07]
    Finbow, S., King, A., MacGillivray, G., Rizzi, R.: The firefighter problem for graphs of maximum degree three. Discrete Mathematics 307(16), 2094–2105 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [FM07]
    Finbow, S., MacGillivray, G.: The firefighter problem: a survey (manuscript, 2007)Google Scholar
  7. [Fog03]
    Fogarty, P.: Catching the Fire on Grids, M.Sc. Thesis, Department of Mathematics, University of Vermont (2003)Google Scholar
  8. [Har06]
    Hartke, S.G.: Attempting to narrow the integrality gap for the firefighter problem on trees. In: Discrete Methods in Epidemiology. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 70, pp. 179–185 (2006)Google Scholar
  9. [Har95]
    Hartnell, B.: Firefighter! An application of domination. In: 24th Manitoba Conference on Combinatorial Mathematics and Computing, University of Minitoba, Winnipeg, Cadada (1995)Google Scholar
  10. [HL00]
    Hartnell, B., Li, Q.: Firefighting on trees: how bad is the greedy algorithm? Congr. Numer. 145, 187–192 (2000)MathSciNetzbMATHGoogle Scholar
  11. [IP01]
    Impagliazzo, R., Paturi, R.: On the complexity of k-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [IPZ01]
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [MW03]
    MacGillivray, G., Wang, P.: On the firefighter problem. J. Combin. Math. Combin. Comput. 47, 83–96 (2003)MathSciNetzbMATHGoogle Scholar
  14. [NSS95]
    Naor, M., Schulman, L.J., Srinivasan, A.: Splitters and near-optimal derandomization. In: IEEE Symposium on Foundations of Computer Science, pp. 182–191 (1995)Google Scholar
  15. [Ver]
    Verbin, E.: Asymmetric universal sets (in preparation)Google Scholar
  16. [WM02]
    Wang, P., Moeller, S.: Fire control on graphs. J. Combin. Math. Combin. Comput. 41, 19–34 (2002)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Leizhen Cai
    • 1
  • Elad Verbin
    • 2
  • Lin Yang
    • 1
  1. 1.Department of Computer Science and EngineeringThe Chinese University of Hong Kong, ShatinHong Kong SARChina
  2. 2.The Institute For Theoretical Computer ScienceTsinghua UniversityBeijingChina

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