Advertisement

Abstract

The interdiction problem arises in a variety of areas including military logistics, infectious disease control, and counter-terrorism. In the typical formulation of network interdiction, the task of the interdictor is to find a set of edges in a weighted network such that the removal of those edges would maximally increase the cost to an evader of traveling on a path through the network.

Our work is motivated by cases in which the evader has incomplete information about the network or lacks planning time or computational power, e.g. when authorities set up roadblocks to catch bank robbers, the criminals do not know all the roadblock locations or the best path to use for their escape.

We introduce a model of network interdiction in which the motion of one or more evaders is described by Markov processes and the evaders are assumed not to react to interdiction decisions. The interdiction objective is to find an edge set of size B, that maximizes the probability of capturing the evaders.

We prove that similar to the standard least-cost formulation for deterministic motion this interdiction problem is also NP-hard. But unlike that problem our interdiction problem is submodular and the optimal solution can be approximated within 1 − 1/e using a greedy algorithm. Additionally, we exploit submodularity through a priority evaluation strategy that eliminates the linear complexity scaling in the number of network edges and speeds up the solution by orders of magnitude. Taken together the results bring closer the goal of finding realistic solutions to the interdiction problem on global-scale networks.

Keywords

Greedy Algorithm Target Node Submodular Function Naval Research Logistics Priority Algorithm 
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.
    Corley, H.W., Sha, D.Y.: Most vital links and nodes in weighted networks. Oper. Res. Lett. 1(4), 157–160 (1982)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    McMasters, A.W., Mustin, T.M.: Optimal interdiction of a supply network. Naval Research Logistics Quarterly 17(3), 261–268 (1970)CrossRefMATHGoogle Scholar
  3. 3.
    Ghare, P.M., Montgomery, D.C., Turner, W.C.: Optimal interdiction policy for a flow network. Naval Research Logistics Quarterly 18(1), 37 (1971)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Pourbohloul, B., Meyers, L., Skowronski, D., Krajden, M., Patrick, D., Brunham, R.: Modeling control strategies of respiratory pathogens. Emerg. Infect. Dis. 11(8), 1246–1256 (2005)CrossRefGoogle Scholar
  5. 5.
    Farley, J.D.: Breaking Al Qaeda cells: A mathematical analysis of counterterrorism operations (a guide for risk assessment and decision making). Studies in Conflict and Terrorism 26, 399–411 (2003)CrossRefGoogle Scholar
  6. 6.
    Pan, F., Charlton, W., Morton, D.P.: Interdicting smuggled nuclear material. In: Woodruff, D. (ed.) Network Interdiction and Stochastic Integer Programming, pp. 1–19. Kluwer Academic Publishers, Boston (2003)Google Scholar
  7. 7.
    Ball, M.O., Golden, B.L., Vohra, R.V.: Finding the most vital arcs in a network. Oper. Res. Lett. 8(2), 73–76 (1989)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bar-Noy, A., Khuller, S., Schieber, B.: The complexity of finding most vital arcs and nodes. Technical report, University of Maryland, College Park, MD, USA (1995)Google Scholar
  9. 9.
    Boros, E., Borys, K., Gurevich, V.: Inapproximability bounds for shortest-path network intediction problems. Technical report, Rutgers University, Piscataway, NJ, USA (2006)Google Scholar
  10. 10.
    Gutfraind, A., Hagberg, A., Izraelevitz, D., Pan, F.: Interdicting a Markovian evader (preprint) (2009)Google Scholar
  11. 11.
    Grinstead, C.M., Snell, J.L.: Introduction to Probability. Second revised edn. American Mathematical Society, USA (July 1997)Google Scholar
  12. 12.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum, New York (1972)CrossRefGoogle Scholar
  13. 13.
    Nemhauser, G., Wolsey, L., Fisher, M.: An analysis of the approximations for maximizing submodular set functions-I. Mathematical Programming 14, 265–294 (1978)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Khuller, S., Moss, A., Naor, J.S.: The budgeted maximum coverage problem. Information Processing Letters 70(1), 39–45 (1999)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Krause, A., Guestrin, C.: A note on the budgeted maximization on submodular functions. Technical report, Carnegie Mellon University, CMU-CALD-05-103 (2005)Google Scholar
  16. 16.
    Leskovec, J., Krause, A., Guestrin, C., Faloutsos, C., VanBriesen, J., Glance, N.: Cost-effective outbreak detection in networks. In: KDD 2007: Proceedings of the 13th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 420–429. ACM, New York (2007)Google Scholar
  17. 17.
    Bradonjić, M., Kong, J.S.: Wireless ad hoc networks with tunable topology. In: Forty-Fifth Annual Allerton Conference, UIUC, Illinois, USA, pp. 1170–1177 (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Alexander Gutfraind
    • 1
  • Aric Hagberg
    • 2
  • Feng Pan
    • 3
  1. 1.Center for Applied MathematicsCornell University, IthacaNew YorkUSA
  2. 2.Theoretical DivisionLos Alamos National LaboratoryLos AlamosUSA
  3. 3.Risk Analysis and Decision Support SystemsLos Alamos National LaboratoryLos AlamosUSA

Personalised recommendations