Abstract
We provide approximation algorithms for several variants of the Firefighter problem on general graphs. The Firefighter problem models the case where an infection or another diffusive process (such as an idea, a computer virus, or a fire) is spreading through a network, and our goal is to stop this infection by using targeted vaccinations. Specifically, we are allowed to vaccinate at most B nodes per time-step (for some budget B), 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 idea. We give complexity and approximation results for problems on both models.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Full version can be found at www.cs.rpi.edu/~eanshel/pubs.html
Ahuja, R., Magnanti, T., Orlin, J.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cliffs (1993)
Aspnes, J., Chang, K., Yamposlkiy, A.: Inoculation strategies for victims of viruses and the sum-of-squares partition problem. In: Proc. 16th ACM SODA (2005)
Bailey, N.: The Mathematical Theory of Infectious Diseases and its Applications. Hafner Press (1975)
Barabasi, A., Albert, R., Jeong, H.: Mean-field theory for scale-free random networks. Physica A 272 (1999)
Calinescu, G., Chekuri, C., Pal, M., Vondrak, J.: Maximizing a Monotone Submodular Function subject to a Matroid Constraint. In: Proc. 12th IPCO (2007)
Chalermsook, P., Chuzhoy, J.: Resource Minimization for Fire Containment. To appear in Proc. ACM SODA (2010)
Crosby, S., Finbow, A., Hartnell, B., Moussi, R., Patterson, K., Wattar, D.: Designing Fire Resistant Graphs. Congr. Numerantium 173 (2005)
Develin, M., Hartke, S.G.: Fire Containment in grids of dimension three or higher. Discrete Applied Mathematics 155(17) (2007)
Dezső, Z., Barabási, A.: Halting viruses in scale-free networks. Physical Review E 65 (2002)
Dreyer Jr., P.A., Roberts, F.S.: Irreversible k-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7) (2009)
Engelberg, R., Könemann, J., Leonardi, S., Naor, J(S.): Cut problems in graphs with a budget constraint. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 435–446. Springer, Heidelberg (2006)
Eubank, S., Kumar, V., Marathe, M., Srinivasan, A., Wang, N.: Structural and algorithmic aspects of massive social networks. In: Proc. 15th ACM SODA (2004)
Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45 (1998)
Finbow, S., King, A.D., MacGillivray, G., Rizzi, R.: The Fire fighter problem on graphs of maximum degree three. Discrete Mathematics 307 (2007)
Finbow, S., MacGillivray, G.: The Firefighter Problem: A survey of results, directions and questions (Manuscript) (2007)
Fisher, M.L., Nemhauser, G.L., Wolsey, L.A.: An analysis of approximations for maximizing submodular set functions - II. Math. Prog. Study 8 (1978)
Fogarty, P.: Catching Fire on Grids, M.Sc. Thesis, Department of Mathematics, University of Vermont (2003)
Giakkoupis, G., Gionis, A., Terzi, E., Tsaparas, P.: Models and algorithms for network immunization. Technical Report C-2005-75, Department of Computer Science, University of Helsinki (2005)
Hartnell, B.L.: Firefighter! An application of domination. Presentation. In: 25th Manitoba Conference on Combinatorial Mathematics and Computing, University of Manitoba in Winnipeg, Canada (1995)
Hartnell, B., Li, Q.: Firefighting on trees: How bad is the greedy algorithm? Congr. Numer. 145 (2000)
Hayrapetyan, A., Kempe, D., Pál, M., Svitkina, Z.: Unbalanced graph cuts. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 191–202. Springer, Heidelberg (2005)
Kempe, D., Kleinberg, J.M., Tardos, É.: Influential nodes in a diffusion model for social networks. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1127–1138. Springer, Heidelberg (2005)
King, A., MacGillivray, G.: The Firefighter Problem For Cubic Graphs. Discrete Mathematics 307 (2007)
Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tomkins, A., Upfal, E.: Stochastic models for the web graph. In: Proc. 41st IEEE FOCS (2000)
Leizhen, C., 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)
Leizhen, C., Weifan, W.: The Surviving Rate of a Graph. To appear in SIAM Journal on Discrete Mathematics (2009)
MacGillivray, G., Wang, P.: On The Firefighter Problem. JCMCC, 47 (2003)
Nowak, M., May, R.: Virus Dynamics: Mathematical Principles of Immunology and Virology. Oxford University Press, Oxford (2000)
Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and sub-constant error-probability PCP characterization of NP. In: Proc. 29th ACM STOC (1997)
Wang, P., Moeller, S.: Fire Control on graphs. J. Combin. Math. Combin. Comput. 41 (2002)
Watts, D., Strogatz, S.: Collective dynamics of ’small-world’ networks. Nature 393 (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Anshelevich, E., Chakrabarty, D., Hate, A., Swamy, C. (2009). Approximation Algorithms for the Firefighter Problem: Cuts over Time and Submodularity. In: Dong, Y., Du, DZ., Ibarra, O. (eds) Algorithms and Computation. ISAAC 2009. Lecture Notes in Computer Science, vol 5878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10631-6_98
Download citation
DOI: https://doi.org/10.1007/978-3-642-10631-6_98
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-10630-9
Online ISBN: 978-3-642-10631-6
eBook Packages: Computer ScienceComputer Science (R0)