Annals of Operations Research

, Volume 249, Issue 1–2, pp 39–53 | Cite as

A simple greedy heuristic for linear assignment interdiction

  • Vladimir Stozhkov
  • Vladimir Boginski
  • Oleg A. Prokopyev
  • Eduardo L. Pasiliao
S.I.: Pardalos60


We consider a bilevel extension of the classical linear assignment problem motivated by network interdiction applications. Specifically, given a bipartite graph with two different (namely, the leader’s and the follower’s) edge costs, the follower solves a linear assignment problem maximizing his/her own profit, whereas the leader is allowed to affect the follower’s decisions by eliminating some of the vertices from the graph. The leader’s objective is to minimize the total cost given by the cost of the interdiction actions plus the cost of the assignments made by the follower. The considered problem is strongly \({ NP}\)-hard. First, we formulate this problem as a linear mixed integer program (MIP), which can be solved by commercial MIP solvers. More importantly, we also describe a greedy-based construction heuristic, which provides (under some mild conditions) an optimal solution for the case, where the leader’s and the follower’s edge costs are equal to one. Finally, we present the results of our computational experiments comparing the proposed heuristic against an MIP solver.


Bilevel programming Assignment interdiction Linear assignment 



This material is based upon work supported by AFRL Mathematical Modeling and Optimization Institute. The research of the first three authors is also partially supported by grants from AFOSR. In addition, we are grateful to Dr. Behdad Beheshti for his helpful comments. Finally, the authors would like to thank anonymous referees whose constructive comments resulted in the improvements to this paper.


  1. Audet, C., Hansen, P., Jaumard, B., & Savard, G. (1997). Links between linear bilevel and mixed 0–1 programming problems. Journal of Optimization Theory and Applications, 93(2), 273–300.CrossRefGoogle Scholar
  2. Bard, J. F. (1998). Practical bilevel optimization. Dordrecht: Kluwer.CrossRefGoogle Scholar
  3. Beheshti, B., Özaltın, O. Y., Zare, M. H., & Prokopyev, O. A. (2015). Exact solution approach for a class of nonlinear bilevel knapsack problems. Journal of Global Optimization, 61(2), 291–310.CrossRefGoogle Scholar
  4. Beheshti, B., Prokopyev, O.A., & Pasiliao, E.L. (2015). Exact solution approach for the bilevel assignment problem. Computational Optimization and Applications, 2015. Accepted for publication.Google Scholar
  5. Brown, G., Carlyle, M., Salmeron, J., & Wood, K. (2006). Defending critical infrastructure. Interfaces, 36, 530–544.CrossRefGoogle Scholar
  6. Burkard, R., Dell’Amico, M., & Martello, S. (2009). Assignment problems. Philadelphia: Society for Industrial Mathematics.CrossRefGoogle Scholar
  7. Colson, B., Marcotte, P., & Savard, G. (2007). An overview of bilevel optimization. Annals of Operations Research, 153(1), 235–256.CrossRefGoogle Scholar
  8. Dempe, S. (2002). Foundations of bilevel programming. Dordrecht: Kluwer.Google Scholar
  9. Deng, X. (1998). Complexity issues in bilevel linear programming. In A. Migdalas, P. M. Pardalos, & P. Varbrand (Eds.), Multilevel optimization: Algorithms and applications (pp. 149–164). Dordrecht: Kluwer.CrossRefGoogle Scholar
  10. Feo, T. A., & Resende, M. G. C. (1995). Greedy randomized adaptive search procedures. Journal of global optimization, 6(2), 109–133.CrossRefGoogle Scholar
  11. Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-completeness. New York: W.H. Freeman.Google Scholar
  12. Gassner, E., & Klinz, B. (2009). The computational complexity of bilevel assignment problems. 4OR, 7(4), 379–394.CrossRefGoogle Scholar
  13. Migdalas, A., Pardalos, P. M., & Värbrand, P. (1998). Multilevel optimization: Algorithms and applications. Norwell: Kluwer.CrossRefGoogle Scholar
  14. Nemhauser, G. L., & Wolsey, L. A. (1988). Integer and Combinatorial Optimization. NewYork, NY: Wiley.CrossRefGoogle Scholar
  15. Resende, M. G. C., & Ribeiro, C. C. (2011). Restart strategies for grasp with path-relinking heuristics. Optimization Letters, 5(3), 467–478.CrossRefGoogle Scholar
  16. Shen, S., Smith, J. C., & Goli, R. (2012). Exact interdiction models and algorithms for disconnecting networks via node deletions. Discrete Optimization, 9(3), 172–188.CrossRefGoogle Scholar
  17. Wood, K. R. (1993). Deterministic network interdiction. Mathematical and Computer Modelling, 17, 1–18.CrossRefGoogle Scholar
  18. Zenklusen, R. (2010). Matching interdiction. Discrete Applied Mathematics, 158(15), 1676–1690.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Vladimir Stozhkov
    • 1
  • Vladimir Boginski
    • 1
    • 4
  • Oleg A. Prokopyev
    • 2
  • Eduardo L. Pasiliao
    • 3
  1. 1.Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA
  2. 2.Department of Industrial EngineeringUniversity of PittsburghPittburghUSA
  3. 3.Munitions DirectorateAir Force Research LaboratoryEglin AFBUSA
  4. 4.Department of Industrial Engineering and Management SystemsUniversity of Central FloridaOrlandoUSA

Personalised recommendations