Skip to main content
SpringerLink
Log in

Exact solution approaches for bilevel assignment problems

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

We consider the bilevel assignment problem in which each decision maker (i.e., the leader and the follower) has its own objective function and controls a distinct set of edges in a given bipartite graph. The leader acts first by choosing some of its edges. Subsequently, the follower completes the assignment process. The edges selected by the leader and the follower are required to constitute a perfect matching. In this paper we propose an exact solution approach, which is based on a branch-and-bound framework and exploits structural properties of the assignment problem. Extensive computational experiments with linear sum and linear bottleneck objective functions are conducted to demonstrate the performance of the developed methods. While the considered problem is known to be NP-hard in general, we also describe some restricted cases that can be solved in polynomial time.

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

Fig. 1

Similar content being viewed by others

References

  1. Aboudi, R., Jørnsten, K.: Resource constrained assignment problems. Discret. Appl. Math. 26(2), 175–191 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aiex, R.M., Resende, M.G.C., Pardalos, P.M., Toraldo, G.: GRASP with path relinking for three-index assignment. INFORMS J. Comput. 17(2), 224–247 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Balinski, M.L., Gomory, R.E.: A primal method for the assignment and transportation problems. Manag. Sci. 10(3), 578–593 (1964)

    Article  Google Scholar 

  4. Beheshti, B.: Test instances for the bilevel assignment problem. http://www.pitt.edu/droleg/files/BAP.html. Accessed April 6, 2014

  5. Boros, E., Hammer, P.L.: Pseudo-boolean optimization. Discret. Appl. Math. 123(1), 155–225 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Burkard, R., Dell’Amico, M., Martello, S.: Assignment Problems. Society for Industrial Mathematics, Philadelphia (2009)

    Book  MATH  Google Scholar 

  7. Chegireddy, C.R., Hamacher, H.W.: Algorithms for finding \(k\)-best perfect matchings. Discret. Appl. Math. 18(2), 155–165 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  8. Colson, B., Marcotte, P., Savard, G.: An overview of bilevel optimization. Ann. Oper. Res. 153(1), 235–256 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Deng, X.: Complexity issues in bilevel linear programming. In: Migdalas, A., Pardalos, P.M., Varbrand, P. (eds.) Multilevel Optimization: Algorithms and Applications, pp. 149–164. Kluwer Academic Publishers, Dordrecht (1998)

    Chapter  Google Scholar 

  10. Fukuda, K., Matsui, T.: Finding all minimum-cost perfect matchings in bipartite graphs. Networks 22(5), 461–468 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Garfinkel, R.S.: An improved algorithm for the bottleneck assignment problem. Oper. Res. 19(7), 1747–1751 (1971)

    Article  MATH  Google Scholar 

  12. Gassner, E., Klinz, B.: The computational complexity of bilevel assignment problems. 4OR Q. J. Oper. Res. 7(4), 379–394 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Glover, F.: Improved linear integer programming formulations of nonlinear integer problems. Manag. Sci. 22(4), 455–460 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  14. King, D.E.: Dlib-ml: a machine learning toolkit. J. Mach. Learn. Res. 10, 1755–1758 (2009)

    Google Scholar 

  15. Krokhmal, P.A.: On optimality of a polynomial algorithm for random linear multidimensional assignment problem. Optim. Lett. 5(1), 153–164 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lieshout, P.M.D., Volgenant, A.: A branch-and-bound algorithm for the singly constrained assignment problem. Eur. J. Oper. Res. 176(1), 151–164 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu, Y.H., Spencer, T.H.: Solving a bilevel linear program when the inner decision maker controls few variables. Eur. J. Oper. Res. 81(3), 644–651 (1995)

    Article  MATH  Google Scholar 

  18. Migdalas, A., Pardalos, P.M., Värbrand, P.: Multilevel Optimization: Algorithms and Applications, vol. 20. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  19. Murty, K.G.: An algorithm for ranking all the assignments in order of increasing cost. Oper. Res. 16(3), 682–687 (1968)

    Article  MATH  Google Scholar 

  20. Pardalos, P.M., Pitsoulis, L.: Nonlinear Assignment Problems: Algorithms and Applications, vol. 7. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  21. Pedersen, C.R., Relund Nielsen, L., Andersen, K.A.: An algorithm for ranking assignments using reoptimization. Comput. Oper. Res. 35(11), 3714–3726 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the Associate Editor and the anonymous reviewers for their constructive comments that helped us to greatly improve the quality of the paper. This material is based upon work supported by the U.S. Air Force Research Laboratory (AFRL) Mathematical Modeling and Optimization Institute and the U.S. Air Force Office of Scientific Research (AFOSR). The research of the second author was also supported by the U.S. Air Force Summer Faculty Fellowship and by AFRL/RW under agreement number FA8651-12-2-0008. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of AFRL/RW or the U.S. Government.

Author information

Authors and Affiliations

  1. Department of Industrial Engineering, University of Pittsburgh, 1048 Benedum Hall, Pittsburgh, PA, 15261, USA

    Behdad Beheshti & Oleg A. Prokopyev

  2. AFRL Munitions Directorate, Building 13, Eglin AFB, FL, 32542, USA

    Eduardo L. Pasiliao

Authors
  1. Behdad Beheshti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oleg A. Prokopyev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eduardo L. Pasiliao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oleg A. Prokopyev.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Beheshti, B., Prokopyev, O.A. & Pasiliao, E.L. Exact solution approaches for bilevel assignment problems. Comput Optim Appl 64, 215–242 (2016). https://doi.org/10.1007/s10589-015-9799-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-015-9799-4

Keywords

Search

Navigation