The Firefighter Problem: Application of Hybrid Ant Colony Optimization Algorithms

  • Christian Blum
  • Maria J. Blesa
  • Carlos García-Martínez
  • Francisco J. Rodríguez
  • Manuel Lozano
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8600)


The firefigther problem is a deterministic discrete-time model for the spread (and the containment) of fire on an undirected graph. Assuming that the fire breaks out at a predefined set of vertices, the goal is to save as many vertices as possible from burning. The same model has also been used in the literature for the simulation of the spreading of deseases. In this work we present, to our knowledge, the first metaheuristics for tackling this problem. In particular, a pure ant colony optimization approach and a hybrid variant of this algorithm are proposed. The results show that the hybrid ant colony optimization variant is superior to the pure ant colony optimization version and to a mathematical programming solver, especially when the graph size and density grows.


Random Graph Heuristic Information Determinism Rate Graph Size Discrete Apply Mathematic 
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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  1. 1.
    Anshelevich, E., Chakrabarty, D., Hate, A., Swamy, C.: Approximation Algorithms for the Firefighter Problem: Cuts over Time and Submodularity. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 974–983. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  2. 2.
    Anshelevich, E., Chakrabarty, D., Hate, A., Swamy, C.: Approximability of the Firefighter Problem. Algorithmica 62(1-2), 520–536 (2010)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bazgan, C., Chopin, M., Ries, B.: The firefighter problem with more than one firefighter on trees. Discrete Applied Mathematics 161(7-8), 899–908 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Blum, C., Dorigo, M.: The hyper-cube framework for ant colony optimization. IEEE Trans. on Man, Systems and Cybernetics – Part B 34(2), 1161–1172 (2004)CrossRefGoogle Scholar
  5. 5.
    Bonato, A., Messinger, M.E., Prałat, P.: Fighting constrained fires in graphs. Theoretical Computer Science 434, 11–22 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Cai, L., Cheng, Y., Verbin, E., Zhou, Y.: Surviving Rates of Graphs with Bounded Treewidth for the Firefighter Problem. SIAM Journal on Discrete Mathematics 24(4), 1322–1335 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    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
  8. 8.
    Cai, L., Wang, W.: The Surviving Rate of a Graph for the Firefighter Problem. SIAM Journal on Discrete Mathematics 23(4), 1814–1826 (2010)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Costa, V., Dantas, S., Dourado, M.C., Penso, L., Rautenbach, D.: More fires and more fighters. Discrete Applied Mathematics 161(16-17), 2410–2419 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Cygan, M., Fomin, F.V., van Leeuwen, E.J.: Parameterized Complexity of Firefighting Revisited. In: Marx, D., Rossmanith, P. (eds.) IPEC 2011. LNCS, vol. 7112, pp. 13–26. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  11. 11.
    Develin, M., Hartke, S.G.: Fire containment in grids of dimension three and higher. Discrete Applied Mathematics 155(17), 2257–2268 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Esperet, L., van den Heuvel, J., Maffray, F., Sipma, F.: Fire Containment in Planar Graphs. Journal of Graph Theory 73(3), 267–279 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Feldheim, O.N., Hod, R.: 3/2 Firefighters Are Not Enough. Discrete Applied Mathematics 161(1-2), 301–306 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Finbow, S., King, A., MacGillivray, G., Rizzi, R.: The firefighter problem for graphs of maximum degree three. Discrete Mathematics 307(16), 2094–2105 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Finbow, S., Science, C., Scotia, N., Macgillivray, G.: The Firefighter Problem: A survey of results, directions and questions. Australian Journal of Combinatorics 43, 57–77 (2009)zbMATHGoogle Scholar
  16. 16.
    Floderus, P., Lingas, A., Persson, M.: Towards more efficient infection and fire fighting. In: CATS 2011 Proceedings of the Seventeenth Computing: The Australasian Theory Symposium, pp. 69–74 (2011)Google Scholar
  17. 17.
    Fogarty, P.: Catching the fire on grids. Master’s thesis, Department of Mathematics. University of Vermont, USA (2003)Google Scholar
  18. 18.
    Fomin, F.V., Heggernes, P., van Leeuwen, E.J.: Making life easier for firefighters. In: Kranakis, E., Krizanc, D., Luccio, F. (eds.) FUN 2012. LNCS, vol. 7288, pp. 177–188. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  19. 19.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer (1988)Google Scholar
  20. 20.
    Hartke, S.G.: Attempting to Narrow the Integrality Gap for the Firefighter Problem on Trees. In: DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pp. 225–231 (2006)Google Scholar
  21. 21.
    Hartnell, B.: Firefighter! An application of domination. In: 20th Conference on Numerical Mathematics and Computing (1995)Google Scholar
  22. 22.
    Hartnell, B., Li, Q.: Firefighting on trees: How bad is the greedy algorithm? In: Proc. of the Thirty-first Southeastern International Conference on Combinatorics, Graph Theory and Computing, pp. 187–192 (2000)Google Scholar
  23. 23.
    Iwaikawa, Y., Kamiyama, N., Matsui, T.: Improved Approximation Algorithms for Firefighter Problem on Trees. IEICE Transactions on Information and Systems E94-D(2), 196–199 (2011)Google Scholar
  24. 24.
    King, A., MacGillivray, G.: The firefighter problem for cubic graphs. Discrete Mathematics 310(3), 614–621 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    MacGillivray, G., Wang, P.: On the firefighter problem. Journal of Combinatorial Mathematics and Combinatorial Computing 47, 83–96 (2003)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Messinger, M.E., Scotia, N.: Firefighting on the Triangular Grid. Journal of Combinatorial Mathematics and Combinatorial Computing 63, 3–45 (2007)MathSciNetGoogle Scholar
  27. 27.
    Messinger, M.E.: Firefighting on Infinite Grids. Master’s thesis, Department of Mathematics and Statistics, Dalhousie University, Halifax, Canada (2004)Google Scholar
  28. 28.
    Moeller, S., Wang, P.: Fire Control on graphs. Journal of Combinatorial Mathematics and Combinatorial Computing 41, 19–34 (2002)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Ng, K., Raff, P.: A generalization of the firefighter problem on. Discrete Applied Mathematics 156(5), 730–745 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Stützle, T., Hoos, H.H.: \({\cal MAX}\)-\({\cal MIN}\) Ant System. Future Generation Computer Systems 16(8), 889–914 (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Christian Blum
    • 1
    • 2
  • Maria J. Blesa
    • 3
  • Carlos García-Martínez
    • 4
  • Francisco J. Rodríguez
    • 5
  • Manuel Lozano
    • 5
  1. 1.Dept. of Computer Science and Artifical IntelligenceUniv. of the Basque Country UPV/EHUSan SebastianSpain
  2. 2.IKERBASQUEBasque Foundation for ScienceBilbaoSpain
  3. 3.ALBCOM Research GroupUniv. Politécnica de CatalunyaBarcelonaSpain
  4. 4.Dept. of Computing and Numerical AnalysisUniv. of CórdobaSpain
  5. 5.Dept. of Computer Science and Artificial IntelligenceUniv. of GranadaSpain

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