Continuous Firefighting on Infinite Square Grids

  • Xujin Chen
  • Xiaodong Hu
  • Changjun Wang
  • Ying Zhang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10185)


The classical firefighter problem, introduced by Bert Hartnell in 1995, is a deterministic discrete-time model of the spread and defence of fire, rumor, or disease. In contrast to the generally “discontinuous” firefighter movements of the classical setting, we propose in the paper the continuous firefighting model. Given an undirected graph G, at time 0, all vertices of G are undefended, and fires break out on one or multiple different vertices of G. At each subsequent time step, the fire spreads from each burning vertex to all of its undefended neighbors. A finite number of firefighters are available to be assigned on some vertices of G at time 1, and each firefighter can only move from his current location (vertex) to one of his neighbors or stay still at each time step. A vertex is defended if some firefighter reaches it no later than the fire. We study fire containment on infinite k-dimensional square grids under the continuous firefighting model. We show that the minimum number of firefighters needed is exactly 2k for single fire, and 5 for multiple fires when \(k=2\).


Firefighter problem Continuous firefighting Fire containment Infinite square grids 


  1. 1.
    Anshelevich, E., Chakrabarty, D., Hate, A., Swamy, C.: Approximation algorithms for the firefighter problem: cuts over time and submodularity. In: Proceedings of the 20th International Symposium on Algorithms and Computation, pp. 974–983 (2009)Google Scholar
  2. 2.
    Cai, L., Verbin, E., Yang, L.: Firefighting on trees: \((1-1/e)\)-approximation, fixed parameter tractability and a subexponential algorithm. In: Proceedings of the 19th International Symposium on Algorithms and Computation, pp. 258–269 (2008)Google Scholar
  3. 3.
    Chalermsook, P., Chuzhoy, J.: Resource minimization for fire containment. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1334–1349 (2010)Google Scholar
  4. 4.
    Comellas, F., Mitjana, M., Peters, J.G.: Epidemics in small world communication networks. Technical report, SFU-CMPT-TR-2002 (2002)Google Scholar
  5. 5.
    Develin, M., Hartke, S.G.: Fire containment in grids of dimension three and higher. Discrete Appl. Math. 155(17), 2257–2268 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dezső, Z., Barabási, A.L.: Halting viruses in scale-free networks. Phys. Rev. E 65(5), 055103 (2002)CrossRefGoogle Scholar
  7. 7.
    Finbow, S., King, A., MacGillivray, G., Rizzi, R.: The firefighter problem for graphs of maximum degree three. Discrete Math. 307(16), 2094–2105 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Finbow, S., MacGillivray, G.: The firefighter problem: a survey of results, directions and questions. Australas. J. Comb. 43, 57–77 (2009)MathSciNetMATHGoogle Scholar
  9. 9.
    Fogarty, P.: Catching the fire on grids. Master’s thesis, The University of Vermont (2003)Google Scholar
  10. 10.
    Hartnell, B.: Firefighter! an application of domination. presentation. In: 25th Manitoba Conference on Combinatorial Mathematics and Computing, University of Manitoba in Winnipeg, Canada (1995)Google Scholar
  11. 11.
    King, A., MacGillivray, G.: The firefighter problem for cubic graphs. Discrete Math. 310(3), 614–621 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    MacGillivray, G., Wang, P.: On the firefighter problem. J. Comb. Math. Comb. Comput. 47, 83–96 (2003)MathSciNetMATHGoogle Scholar
  13. 13.
    Messinger, M.E.: Firefighting on infinite grids. Master’s thesis, Department of Mathematics and Statistics, Dalhousie University (2005)Google Scholar
  14. 14.
    Messinger, M.E.: Average firefighting on infinite grids. Australas. J. Comb. 41, 15 (2008)MathSciNetMATHGoogle Scholar
  15. 15.
    Ng, K., Raff, P.: Fractional firefighting in the two dimensional grid, pp. 2005–23. Technical report, DIMACS (2005)Google Scholar
  16. 16.
    Scott, A.E., Stege, U., Zeh, N.: Politicians firefighting. In: Proceedings of the 17th International Symposium on Algorithms and Computation, pp. 608–617 (2006)Google Scholar
  17. 17.
    Wang, P., Moeller, S.A.: Fire control on graphs. J. Comb. Math. Comb. Comput. 41, 19–34 (2002)MathSciNetMATHGoogle Scholar
  18. 18.
    Yutaka, I., Naoyuki, K., Tomomi, M.: Improved approximation algorithms for firefighter problem on trees. IEICE Trans. Inf. Syst. 94(2), 196–199 (2013)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Xujin Chen
    • 1
  • Xiaodong Hu
    • 1
  • Changjun Wang
    • 2
  • Ying Zhang
    • 1
  1. 1.Academy of Mathematics and Systems Science, Chinese Academy of SciencesBeijingChina
  2. 2.Beijing Institute for Scientific and Engineering ComputingBeijing University of TechnologyBeijingChina

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