Advertisement

Approximation Algorithms for the Geometric Firefighter and Budget Fence Problems

  • Rolf Klein
  • Christos Levcopoulos
  • Andrzej Lingas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)

Abstract

Let R denote a connected region inside a simple polygon, P. By building 1-dimensional barriers in P ∖ R, we want to separate from R part(s) of P of maximum area. In this paper we introduce two versions of this problem. In the budget fence version the region R is static, and there is an upper bound on the total length of barriers we may build. In the basic geometric firefighter version we assume that R represents a fire that is spreading over P at constant speed (varying speed can also be handled). Building a barrier takes time proportional to its length, and each barrier must be completed before the fire arrives. In this paper we are assuming that barriers are chosen from a given set B that satisfies a certain linearity condition. For example, this condition is satisfied for barrier curves in general position, if any two barriers cross at most once.

Even for simple cases (e. g., P a convex polygon and B the set of all diagonals), both problems are shown to be NP-hard. Our main result is an efficient ≈ 11.65 approximation algorithm for the firefighter problem. Since this algorithm solves a much more general problem—a hybrid of scheduling and maximum coverage—it may find wider application. We also provide a polynomial-time approximation scheme for the budget fence problem, for the case where barriers chosen from B must not cross.

Keywords

Approximation Algorithm Convex Polygon Simple Polygon Constant Approxi Algebraic Degree 
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.
    Altshuler, Y., Bruckstein, A.M.: On Short Cuts or Fencing in Rectangular Strips. arXiv:1911.5920v1[cs.CG] (November 26, 2010)Google Scholar
  2. 2.
    Anshelevich, E., Chakrabarty, D., Hate, A., Swamy, C.: Approximability of the Firefighter Problem: Computing Cuts over Time. Algorithmica 62(1-2), 520–536 (2012), preliminary version in proc. ISAAC 2009Google Scholar
  3. 3.
    Bansal, N., Gupta, A., Krishnaswamy, R.: A Constant Factor Approximation Algorithm for Generalized Min-Sum Set Cover. In: Proc. SODA, pp. 1539–1545 (2010)Google Scholar
  4. 4.
    Bansal, N., Pruhs, K.: The Geometry of Scheduling. In: FOCS 2010, pp. 407–414 (2010)Google Scholar
  5. 5.
    Barghi, A., Winkler, P.: Firefighting on a random geometric graph. Random Structures & Algorithms, doi:10.1002/rsa.20511 (first published online: June 27, 2013)Google Scholar
  6. 6.
    Cabello, S., Giannopoulos, P.: The Complexity of Separating Points in the Plane. In: Proc. 29th ACM Symposium on Computational Geometry, pp. 379–386 (2013)Google 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.
    Chekuri, C., Vondrák, R., Zenklusen, R.: Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes. Prel. version in STOC 2011, 783–792 (2011), Revised version in http://arxiv.org/pdf/1105.4593v3.pdf (July 30, 2012)
  9. 9.
    Cohen, R., Katzir, L.: The Generalized Maximum Coverage Problem. Information Processing Letters 108, 15–22 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    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
  12. 12.
    Finbow, S., MacGillivray, G.: The Firefighter Problem: a survey of results, directions and questions. Australasian J. Comb. 43, 57–78 (2009)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Floderus, P., Lingas, A., Persson, M.: Towards more efficient infection and fire fighting. Int. J. Found. Comput. Sci. 24(1), 3–14 (2013), preliminary version in proc. CATS 2011Google Scholar
  14. 14.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of NP-completeness. W.H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  15. 15.
    Ghaderi, R., Esnaashari, M., Meybodi, M.R.: An Adaptive Scheduling Algorithm for Set Cover Problem in Wireless Sensor Networks: A Cellular Learning Automata Approach. International Journal of Machine Learning and Computing 2(5) (October 2012)Google Scholar
  16. 16.
    Hassin, R., Levin, A.: An Approximation Algorithm for the Minimum Latency Set Cover Problem. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 726–733. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. 17.
    Khuller, S., Moss, A., Naor, J.S.: The budgeted maximum coverage problem. Information Processing Letters 70, 39–45 (1999)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Rolf Klein
    • 1
  • Christos Levcopoulos
    • 2
  • Andrzej Lingas
    • 2
  1. 1.Institut für Informatik IUniversität BonnBonnGermany
  2. 2.Department of Computer ScienceLund UniversityLundSweden

Personalised recommendations