Advertisement

Parameterized Complexity of Firefighting Revisited

  • Marek Cygan
  • Fedor V. Fomin
  • Erik Jan van Leeuwen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7112)

Abstract

The Firefighter problem is to place firefighters on the vertices of a graph to prevent a fire with known starting point from lighting up the entire graph. In each time step, a firefighter may be permanently placed on an unburned vertex and the fire spreads to its neighborhood in the graph in so far no firefighters are protecting those vertices. The goal is to let as few vertices burn as possible. This problem is known to be NP-complete, even when restricted to bipartite graphs or to trees of maximum degree three. Initial study showed the Firefighter problem to be fixed-parameter tractable on trees in various parameterizations. We complete these results by showing that the problem is in FPT on general graphs when parameterized by the number of burned vertices, but has no polynomial kernel on trees, resolving an open problem. Conversely, we show that the problem is W[1]-hard when parameterized by the number of unburned vertices, even on bipartite graphs. For both parameterizations, we additionally give refined algorithms on trees, improving on the running times of the known algorithms.

Keywords

Bipartite Graph Planar Graph Maximum Degree Parameterized Complexity General Graph 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12, 308–340 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bazgan, C., Chopin, M., Fellows, M.R.: Parameterized Complexity of the Firefighter Problem. In: Asano, T., Nakano, S.-i., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 643–652. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Computing 25(6), 1305–1317 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Cross-composition: A new technique for kernelization lower bounds. CoRR abs/1011.4224 (2010)Google Scholar
  6. 6.
    Cai, L., Verbin, E., Yang, L.: Firefighting on Trees (1-1/e)-Approximation, Fixed Parameter Tractability and a Subexponential Algorithm. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 258–269. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  8. 8.
    Eppstein, D.: Diameter and treewidth in minor-closed graph families. Algorithmica 27(3), 275–291 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Finbow, S., Hartnell, B., Li, Q., Schmeisser, K.: On minimizing the effects of fire or a virus on a network. J. Combin. Math. Combin. Comput. 33, 311–322 (2000)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Finbow, S., King, A., MacGillivray, G., Rizzi, R.: The firefighter problem for graphs of maximum degree three. Discrete Math. 307(16), 2094–2105 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Finbow, S., MacGillivray, G.: The firefighter problem: a survey of results, directions and questions. Australas. J. Combin. 43, 57–77 (2009)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Floderus, P., Lingas, A., Persson, M.: Towards more efficient infection and fire fighting. In: CATS 2011: 17th Computing: The Australasian Theory Symposium. CRPIT, vol. 119, pp. 69–74. Australian Computer Society (2011)Google Scholar
  13. 13.
    Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. In: STOC 2008: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, pp. 133–142. ACM (2008)Google Scholar
  14. 14.
    MacGillivray, G., Wang, P.: On the firefighter problem. J. Combin. Math. Combin. Comput. 47, 83–96 (2003)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Yang, L.: Efficient Algorithms on Trees. M. Phil thesis, Department of Computer Science and Engineering. The Chinese University of Hong Kong (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Marek Cygan
    • 1
  • Fedor V. Fomin
    • 2
  • Erik Jan van Leeuwen
    • 2
  1. 1.Institute of InformaticsUniversity of WarsawPoland
  2. 2.Department of InformaticsUniversity of BergenNorway

Personalised recommendations