Skip to main content

Advertisement

SpringerLink
Log in

Convex hull representation of the deterministic bipartite network interdiction problem

  • Full Length Paper
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

We consider a version of the stochastic network interdiction problem modeled by Morton et al. (IIE Trans 39:3–14, 2007) in which an interdictor attempts to minimize a potential smuggler’s chance of evasion via discrete deployment of sensors on arcs in a bipartite network. The smuggler reacts to sensor deployments by solving a maximum-reliability path problem on the resulting network. In this paper, we develop the (minimal) convex hull representation for the polytope linking the interdictor’s decision variables with the smuggler’s for the case in which the smuggler’s origin and destination are known and interdictions are cardinality-constrained. In the process, we propose an exponential class of easily-separable inequalities that generalize all of those developed so far for the bipartite version of this problem. We show how these cuts may be employed in a cutting-plane fashion when solving the more difficult problem in which the smuggler’s origin and destination are stochastic, and argue that some instances of the stochastic model have facets corresponding to the solution of NP-hard problems. Our computational results show that the cutting planes developed in this paper may strengthen the linear programming relaxation of the stochastic model by as much as 25 %.

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

Access this article

Log in via an institution

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. Atamtürk, A., Nemhauser, G.L., Savelsbergh, M.W.P.: The mixed vertex packing problem. Math. Program. 89, 35–53 (2000)

    MATH  MathSciNet  Google Scholar 

  2. Bailey, M.D., Shechter, S.M., Schaefer, A.J.: SPAR: stochastic programming with adversarial recourse. Oper. Res. Lett. 34, 307–315 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0–1 programs. Math. Program. 58, 295–324 (1993)

    Article  MATH  Google Scholar 

  4. Ball, M.O., Golden, B.L., Vohra, R.V.: Finding the most vital arcs in a network. Oper. Res. Lett. 8, 73–76 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bayrak, H., Bailey, M.D.: Shortest path network interdiction with asymmetric information. Networks 52, 133–140 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Corley, H.W., Chang, H.: Finding the \(n\) most vital nodes in a flow network. Manag. Sci. 21, 362–364 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  7. Corley, H.W., Sha, D.Y.: Most vital links and nodes in weighted networks. Oper. Res. Lett. 1, 157–160 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cormican, K.J., Morton, D.P., Wood, R.K.: Stochastic network interdiction. Oper. Res. 46, 184–197 (1998)

    Article  MATH  Google Scholar 

  9. Fulkerson, D., Harding, G.: Maximizing the minimum source-sink path subject to a budget constraint. Math. Program. 13, 116–118 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  10. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. W. H. Freeman, San Francisco (1979)

    MATH  Google Scholar 

  11. Ghare, P.M., Montgomery, D.C., Turner, W.C.: Optimal interdiction policy for a flow network. Nav. Res. Logist. 18, 37–45 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  12. Golden, B.: A problem in network interdiction. Nav. Res. Logist. 25, 711–713 (1978)

    Article  MATH  Google Scholar 

  13. Günlük, O., Pochet, Y.: Mixing mixed-integer inequalities. Math. Program. 90, 429–457 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Israeli, E., Wood, R.K.: Shortest-path network interdiction. Networks 40, 97–111 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Janjarassuk, U., Linderoth, J.T.: Reformulation and sampling to solve a stochastic network interdiction problem. Networks 52, 120–132 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lim, C., Smith, J.C.: Algorithms for discrete and continuous multicommodity flow network interdiction problems. IIE Trans. 39, 15–26 (2007)

    Article  Google Scholar 

  17. Lovász, L., Schrijver, A.: Cones of matrices and setfunctions and 0–1 optimization. SIAM J. Optim. 1, 166–190 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  18. Lubore, S.H., Ratliff, H.D., Sicilia, G.T.: Determining the most vital link in a flow network. Nav. Res. Logist. 18, 497–502 (1971)

    Article  MathSciNet  Google Scholar 

  19. Malik, K., Mittal, A., Gupta, S.: The \(k\) most vital arcs in the shortest path problem. Oper. Res. Lett. 8, 223–227 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  20. McMasters, A., Mustin, T.: Optimal interdiction of a supply network. Nav. Res. Logist. 17, 261–268 (1970)

    Article  MATH  Google Scholar 

  21. Miller, A.J., Wolsey, L.A.: Tight formulations for some simple mixed integer programs and convex objective integer programs. Math. Program. 98, 73–88 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  22. Morton, D.P., Pan, F., Saeger, K.J.: Models for nuclear smuggling interdiction. IIE Trans. 39, 3–14 (2007)

    Google Scholar 

  23. Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization, 2nd edn. Wiley, New York (1999)

    MATH  Google Scholar 

  24. Pan, F.: Stochastic Network Interdiction: Models and Methods. PhD thesis, University of Texas at Austin (2005)

  25. Pan, F., Morton, D.P.: Minimizing a stochastic maximum-reliability path. Networks 52, 111–119 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. Ratliff, H.D., Sicilia, G.T., Lubore, S.H.: Finding the \(n\) most vital links in flow networks. Manag. Sci. 21, 531–539 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  27. Salmerón, J.: Deception tactics for network interdiction: a multi-objective approach. Networks 60, 45–58 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  28. Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discret. Math. 3(3), 411–430 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  29. Sherali, H.D., Adams, W.P.: A hierarchy of relaxations and convex hull characterizations for mixed-integer zero-one programming problems. Discret. Appl. Math. 52(1), 83–106 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  30. Sherali, H.D., Adams, W.P., Driscoll, P.J.: Exploiting special structures in constructing a hierarchy of relaxations for 0–1 mixed integer programs. Oper. Res. 46, 396–405 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  31. Smith, J.C., Lim, C., Sudargho, F.: Survivable network design under optimal and heuristic interdiction scenarios. J. Glob. Optim. 38, 181–199 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  32. Sullivan, K.M., Morton, D.P., Pan, F., Smith, J.C.: Securing a Border under Asymmetric Information. Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, Technical report (2011)

  33. Wollmer, R.D.: Removing arcs from a network. Oper. Res. 12, 934–940 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  34. Wollmer, R.D.: Algorithms for targeting strikes in a lines-of-communication network. Oper. Res. 18, 497–515 (1970)

    Article  Google Scholar 

  35. Wolsey, L.A.: Integer Programming. Wiley-Interscience, New York NY (1998)

    MATH  Google Scholar 

  36. Wood, R.K.: Deterministic network interdiction. Math. Comput. Model. 17, 1–18 (1993)

    MATH  Google Scholar 

Download references

Acknowledgments

The authors are grateful for the comments of two anonymous referees and an associate editor, whose remarks helped to significantly improve the clarity of our exposition. This work has been supported by the National Science Foundation through grants CMMI-0800676 and CMMI-1100765, the Defense Threat Reduction Agency through grants HDTRA1-08-1-0029, and HDTRA1-10-1-0050, and the US Department of Homeland Security under Grant Award Number 2008-DN-077-ARI021-05. The views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of the US Department of Homeland Security.

Author information

Authors and Affiliations

  1. Department of Industrial Engineering, University of Arkansas, 4207 Bell Engineering Center, Fayetteville, AR, 72701, USA

    Kelly M. Sullivan

  2. Department of Industrial and Systems Engineering, University of Florida, 303 Weil Hall, P.O. Box 116595, Gainesville, FL, 32611-6595, USA

    J. Cole Smith

  3. Graduate Program in Operations Research and Industrial Engineering, The University of Texas at Austin, Austin, TX, 78712-0292, USA

    David P. Morton

Authors
  1. Kelly M. Sullivan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Cole Smith
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David P. Morton
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Cole Smith.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sullivan, K.M., Smith, J.C. & Morton, D.P. Convex hull representation of the deterministic bipartite network interdiction problem. Math. Program. 145, 349–376 (2014). https://doi.org/10.1007/s10107-013-0650-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-013-0650-3

Mathematics Subject Classification

Search

Navigation