Firefighting as a Game

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8882)


The Firefighter Problem was proposed in 1995 [16] as a deterministic discrete-time model for the spread (and containment) of a fire. Its applications reach from real fires to the spreading of diseases and the containment of floods. Furthermore, it can be used to model the spread of computer viruses or viral marketing in communication networks.

In this work, we study the problem from a game-theoretical perspective. Such a context seems very appropriate when applied to large networks, where entities may act and make decisions based on their own interests, without global coordination.

We model the Firefighter Problem as a strategic game where there is one player for each time step who decides where to place the firefighters. We show that the Price of Anarchy is linear in the general case, but at most 2 for trees. We prove that the quality of the equilibria improves when allowing coalitional cooperation among players. In general, we have that the Price of Anarchy is in \(\Theta (\frac{n}{k})\) where \(k\) is the coalition size. Furthermore, we show that there are topologies which have a constant Price of Anarchy even when constant sized coalitions are considered.


Firefighter problem Spreading models for networks Algorithmic game theory Nash equilibria Price of anarchy Coalitions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alon, N., Feldman, M., Procaccia, A.D., Tennenholtz, M.: A note on competitive diffusion through social networks. Information Processing Letters 110, 221–225 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Àlvarez, C., Blesa, M., Molter, H.: Firefighting as a Game. Technical Report LSI-14-9-R, Computer Science Dept, Universitat Politècnica de Catalunya (2014)Google Scholar
  3. 3.
    Anshelevich, E., Chakrabarty, D., Hate, A., Swamy, C.: Approximability of the firefighter problem. Algorithmica 62, 520–536 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bazgan, C., Chopin, M., Cygan, M., Fellows, M.R., Fomin, F., Jan van Leeuwen, E.: Parameterized complexity of firefighting. Journal of Computer and System Sciences 80, 1285–1297 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bazgan, C., Chopin, M., Ries, B.: The firefighter problem with more than one firefighter on trees. Discrete Applied Mathematics 161, 899–908 (2013)CrossRefzbMATHMathSciNetGoogle 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.
    Chlebíková, J., Chopin, M.: The firefighter problem: A structural analysis. Electronic Colloquium on Computational Complexity 20, 162 (2013)Google Scholar
  8. 8.
    Costa, V., Dantas, S., Dourado, M.C., Penso, L., Rautenbach, D.: More fires and more fighters. Discrete Applied Mathematics 161, 2410–2419 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    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
  10. 10.
    Feldheim, O.N., Hod, R.: 3/2 Firefighters Are Not Enough. Discrete Applied Mathematics 161, 301–306 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Finbow, S., King, A., MacGillivray, G., Rizzi, R.: The firefighter problem for graphs of maximum degree three. Discrete Mathematics 307, 2094–2105 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Finbow, S., MacGillivray, G.: The Firefighter Problem: A survey of results, directions and questions. Australian Journal of Combinatorics 43, 57–77 (2009)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Floderus, P., Lingas, A., Persson, M.: Towards more efficient infection and fire fighting. International Journal of Foundations of Computer Science 24, 3–14 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    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
  15. 15.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer (1988)Google Scholar
  16. 16.
    Hartnell, B.: Firefighter! an application of domination. In: 25th Manitoba Conference on Combinatorial Mathematics and Computing, University of Manitoba in Winnipeg, Canada (1995)Google Scholar
  17. 17.
    Hartnell, B., Li, Q.: Firefighting on trees: How bad is the greedy algorithm? Congressus Numerantium 145, 187–192 (2000)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Iwaikawa, Y., Kamiyama, N., Matsui, T.: Improved Approximation Algorithms for Firefighter Problem on Trees. IEICE Transactions 94-D, 196–199 (2011)Google Scholar
  19. 19.
    King, A., MacGillivray, G.: The firefighter problem for cubic graphs. Discrete Mathematics 310, 614–621 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    MacGillivray, G., Wang, P.: On the firefighter problem. Journal of Combinatorial Mathematics and Combinatorial Computing 47, 83–96 (2003)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Ng, K., Raff, P.: A generalization of the firefighter problem on Z \(\times \) Z. Discrete Applied Mathematics 156, 730–745 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Small, L., Mason, O.: Nash Equilibria for competitive information diffusion on trees. Information Processing Letters 113, 217–219 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Small, L., Mason, O.: Information diffusion on the iterated local transitivity model of online social networks. Discrete Applied Mathematics 161, 1338–1344 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Takehara, R., Hachimori, M., Shigeno, M.: A comment on pure-strategy Nash equilibria in competitive diffusion games. Information Processing Letters 112, 59–60 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Zinoviev, D., Duong, V., Zhang, H.: A Game Theoretical Approach to Modeling Information Dissemination in Social Networks. CoRR, abs/1006.5493 (2010)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.ALBCOM Research Group, Computer Science DepartmentUniversitat Politècnica de Catalunya, BarcelonaTechBarcelonaSpain

Personalised recommendations