Geometric Firefighting in the Half-Plane

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11646)


In 2006, Alberto Bressan [3] suggested the following problem. Suppose a circular fire spreads in the Euclidean plane at unit speed. The task is to build, in real time, barrier curves to contain the fire. At each time t the total length of all barriers built so far must not exceed \(t \cdot v\), where v is a speed constant. How large a speed v is needed? He proved that speed \(v>2\) is sufficient, and that \(v>1\) is necessary. This gap of (1, 2] is still open. The crucial question seems to be the following. When trying to contain a fire, should one build, at maximum speed, the enclosing barrier, or does it make sense to spend some time on placing extra delaying barriers in the fire’s way? We study the situation where the fire must be contained in the upper \(L_1\) half-plane by an infinite horizontal barrier to which vertical line segments may be attached as delaying barriers. Surprisingly, such delaying barriers are helpful when properly placed. We prove that speed \(v=1.8772\) is sufficient, while \(v >1.66\) is necessary.


Barrier Firefighting Geodesic circle 



We thank the anonymous referees for their valuable input.


  1. 1.
    Berger, F., Gilbers, A., Grüne, A., Klein, R.: How many lions are needed to clear a grid? Algorithms 2(3), 1069–1086 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Brass, P., Kim, K.D., Na, H.S., Shin, C.S.: Escaping offline searchers and isoperimetric theorems. Comput. Geom. 42(2), 119–126 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bressan, A.: Differential inclusions and the control of forest fires. J. Diff. Eqn. 243(2), 179–207 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bressan, A.: Price offered for a dynamic blocking problem (2011).
  5. 5.
    Bressan, A., Burago, M., Friend, A., Jou, J.: Blocking strategies for a fire control problem. Anal. Appl. 6(3), 229–246 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bressan, A., Wang, T.: The minimum speed for a blocking problem on the half plane. J. Math. Anal. Appl. 356(1), 133–144 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bressan, A., Wang, T.: On the optimal strategy for an isotropic blocking problem. Calc. Var. PDE 45, 125–145 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dumitrescu, A., Suzuki, I., Żyliński, P.: Offline variants of the “lion and man” problem. Theor. Comput. Sci. 399(3), 220–235 (2008)CrossRefGoogle Scholar
  9. 9.
    Finbow, S., King, A., MacGillivray, G., Rizzi, R.: The firefighter problem for graphs of maximum degree three. Discrete Math. 307(16), 2094–2105 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Finbow, S., MacGillivray, G.: The firefighter problem: a survey of results, directions and questions. Technical report (2007)Google Scholar
  11. 11.
    Fomin, F.V., Heggernes, P., van Leeuwen, E.J.: The firefighter problem on graph classes. Theor. Comput. Sci. 613(C), 38–50 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kim, S.S., Klein, R., Kübel, D., Langetepe, E., Schwarzwald, B.: Geometric firefighting in the half-plane. CoRR abs/1905.02067 (2019).
  13. 13.
    Klein, R.: Reversibility properties of the fire-fighting problem in graphs. Comput. Geom. 67, 38–41 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Klein, R., Kübel, D., Langetepe, E., Schwarzwald, B.: Protecting a highway from fire. In: Abstracts EuroCG 2018 (2018)Google Scholar
  15. 15.
    Klein, R., Langetepe, E.: Computational geometry column 63. SIGACT News 47(2), 34–39 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Klein, R., Langetepe, E., Levcopoulos, C.: A fire-fighter’s problem. In: Proceedings 31st Symposium on Computational Geometry (SoCG 2015) (2015)Google Scholar
  17. 17.
    Klein, R., Langetepe, E., Levcopoulos, C., Lingas, A., Schwarzwald, B.: On a fire fighter’s problem. Int. J, Found. Comput. Sci. (2018, to appear)Google Scholar
  18. 18.
    Klein, R., Levcopoulos, C., Lingas, A.: Approximation algorithms for the geometric firefighter and budget fence problem. Algorithms 11, 45 (2018)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of BonnBonnGermany

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