Skip to main content

Advertisement

Log in

Approximability of the Firefighter Problem

Computing Cuts over Time

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We provide approximation algorithms for several variants of the Firefighter problem on general graphs. The Firefighter problem models the case where a diffusive process such as an infection (or an idea, a computer virus, a fire) is spreading through a network, and our goal is to contain this infection by using targeted vaccinations. Specifically, we are allowed to vaccinate at most a fixed number (called the budget) of nodes per time step, with the goal of minimizing the effect of the infection. The difficulty of this problem comes from its temporal component, since we must choose nodes to vaccinate at every time step while the infection is spreading through the network, leading to notions of “cuts over time”.

We consider two versions of the Firefighter problem: a “non-spreading” model, where vaccinating a node means only that this node cannot be infected; and a “spreading” model where the vaccination itself is an infectious process, such as in the case where the infection is a harmful idea, and the vaccine to it is another infectious beneficial idea. We look at two measures: the MaxSave measure in which we want to maximize the number of nodes which are not infected given a fixed budget, and the MinBudget measure, in which we are given a set of nodes which we have to save and the goal is to minimize the budget. We study the approximability of these problems in both models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ahuja, R., Magnanti, T., Orlin, J.: Network Flows: Theory, Algorithms, and Applications. Prentice Hall, New York (1993)

    MATH  Google Scholar 

  2. Aspnes, J., Chang, K., Yamposlkiy, A.: Inoculation strategies for victims of viruses and the sum-of-squares partition problem. J. Comput. Syst. Sci. 72(6), 1077–1093 (2006)

    Article  MATH  Google Scholar 

  3. Baier, G., Erlebach, T., Hall, A., Köhler, E., Schilling, H., Skutella, M.: Length-bounded cuts and flows. Autom. Lang. Program. 4051, 679–690 (2006)

    Article  Google Scholar 

  4. Bailey, N.: The Mathematical Theory of Infectious Diseases and its Applications. Hafner, New York (1975)

    MATH  Google Scholar 

  5. Barabasi, A., Albert, R., Jeong, H.: Mean-field theory for scale-free random networks. Phys. A, Stat. Mech. Appl. 272, 173–187 (1999)

    Article  Google Scholar 

  6. Berger, N., Borgs, C., Chayes, J., Saberi, A.: On the spread of viruses on the internet. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 301–310 (2005)

    Google Scholar 

  7. Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. In: Integer Programming and Combinatorial Optimization, 12th International IPCO Conference, Ithaca, NY, USA (2007)

    Google Scholar 

  8. Chalermsook, P., Chuzhoy, J.: Resource minimization for fire containment. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1334–1349 (2010)

    Google Scholar 

  9. Chekuri, C., Kumar, A.: Maximum coverage problem with group budget constraints and applications. In: Approximation, Randomization, and Combinatorial Optimization. LNCS, pp. 72–83. Springer, Berlin (2004)

    Chapter  Google Scholar 

  10. Develin, M., Hartke, S.G.: Fire Containment in grids of dimension three or higher. Discrete Appl. Math. 155(17), 2257–2268 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Dezső, Z., Barabási, A.: Halting viruses in scale-free networks. Phys. Rev. E 65(5), 055103 (2002)

    Article  Google Scholar 

  12. Dreyer, P. Jr., Roberts, F.: Irreversible k-threshold processes: graph-theoretical threshold models of the spread of disease and of opinion. Discrete Appl. Math. 157(7), 1615–1627 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Engelberg, R., Könemann, J., Leonardi, S., Naor, J.: Cut problems in graphs with a budget constraint. J. Discrete Algorithms 5(2), 262–279 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Eubank, S., Kumar, V., Marathe, M., Srinivasan, A., Wang, N.: Structural and algorithmic aspects of massive social networks. In: Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 711–720 (2004)

    Google Scholar 

  15. Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45, 634–652 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Finbow, S., King, A.D., MacGillivray, Gary, Rizzi, R.: The fire fighter problem on graphs of maximum degree three. Discrete Math. 307, 2094–2105 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Finbow, S., MacGillivray, G.: The firefighter problem: a survey of results, directions and questions. Manuscript (2007)

  18. Fogarty, P.: Catching fire on grids. M.Sc. thesis, Department of Mathematics, University of Vermont (2003)

  19. Ganesh, A., Massoulie, L., Towlsey, D.: The effect of network topology on the spread of epidemics. In: Proc. 24th IEEE INFOCOM Conference, vol. 2, pp. 1455–1466 (2005)

    Google Scholar 

  20. 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 

  21. Hartnell, B., Li, Q.: Firefighting on trees: how bad is the greedy algorithm? Congr. Numer. 145, 187–192 (2000)

    MATH  MathSciNet  Google Scholar 

  22. Hayrapetyan, A., Kempe, D., Pal, M., Svitkina, Z.: Unbalanced graph cuts. In: Proceedings of European Symposium on Algorithms (ESA) (2005)

    Google Scholar 

  23. Kempe, D., Kleinberg, J., Tardos, É.: Influential nodes in a diffusion model for social networks. In: Proceedings of the 32nd International Colloquium on Automata, Languages and Programming (ICALP) (2005)

    Google Scholar 

  24. Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tomkins, A., Upfal, E.: Stochastic models for the web graph. In: Proc. 41st IEEE Symp. on Foundations of Computer Science (FOCS), pp. 57–65 (2000)

    Chapter  Google Scholar 

  25. Cai, L., Verbin, E., Yang, L.: Firefighting on trees: (1−1/e) approximation, fixed parameter tractability and a subexponential algorithm. In: Proceedings of International Symposium on Algorithms and Computation 2008, pp. 258–269 (2008)

    Google Scholar 

  26. MacGillivray, G., Wang, P.: On the firefighter problem. J. Comb. Math. Comb. Comput. 47, 83–96 (2003)

    MATH  MathSciNet  Google Scholar 

  27. Moore, C., Newman, M.: Epidemics and percolation in small-world networks. Phys. Rev. E 61, 5678–5682 (2000)

    Article  Google Scholar 

  28. Nemhauser, G., Wolsey, L.: Integer and Combinatorial Optimization. Wiley, New York (1988)

    MATH  Google Scholar 

  29. Nowak, M., May, R.: Virus Dynamics: Mathematical Principles of Immunology and Virology. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  30. Pastor-Satorras, R., Vespignani, A.: Epidemic dynamics in finite scale-free networks. Phys. Rev. E 65, 035108(R) (2002)

    Google Scholar 

  31. Wang, P., Moeller, S.: Fire control on graphs. J. Comb. Math. Comb. Comput. 41, 19–34 (2002)

    MATH  MathSciNet  Google Scholar 

  32. Watts, D., Strogatz, S.: Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Elliot Anshelevich.

Additional information

A preliminary version of this paper appeared in ISAAC 2009.

Supported in part by NSF CCF-0914782.

Supported in part by NSERC grant 327620-09 and an Ontario Early Researcher Award.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Anshelevich, E., Chakrabarty, D., Hate, A. et al. Approximability of the Firefighter Problem. Algorithmica 62, 520–536 (2012). https://doi.org/10.1007/s00453-010-9469-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-010-9469-y

Keywords

Navigation