Advertisement

4OR

, 7:379 | Cite as

The computational complexity of bilevel assignment problems

  • Elisabeth Gassner
  • Bettina Klinz
Research Paper

Abstract

In bilevel optimization problems there are two decision makers, the leader and the follower, who act in a hierarchy. Each decision maker has his own objective function, but there are common constraints. This paper deals with bilevel assignment problems where each decision maker controls a subset of edges and each edge has a leader’s and a follower’s weight. The edges selected by the leader and by the follower need to form a perfect matching. The task is to determine which edges the leader should choose such that his objective value which depends on the follower’s optimal reaction is maximized. We consider sum- and bottleneck objective functions for the leader and follower. Moreover, if not all optimal reactions of the follower lead to the same leader’s objective value, then the follower either chooses an optimal reaction which is best (optimistic rule) or worst (pessimistic rule) for the leader. We show that all the variants arising if the leader’s and follower’s objective functions are sum or bottleneck functions are NP-hard if the pessimistic rule is applied. In case of the optimistic rule the problem is shown to be NP-hard if at least one of the decision makers has a sum objective function.

Keywords

Bilevel Combinatorial optimization Assignment problem Computational complexity 

MSC classification (2000)

90C27 68Q25 

References

  1. Anandalingam G, Friesz TL (eds) (1992) Hierarchical optimization. Annals of operations research, vol 34. J.C. Baltzer Scientific Publishing Company, BaselGoogle Scholar
  2. Ben-Ayed O, Blair CE (1990) Computational difficulties of bilevel linear programming. Oper Res 38: 556–560CrossRefGoogle Scholar
  3. Burkard RE, Çela E (1999) Linear assignment problems and extensions. In: Du DZ, Pardalos PM (eds) Handbook ofcombinatorial optimization—supplement, vol A. Kluwer, Dordrecht pp 75–149Google Scholar
  4. Burkard RE, Dell’Amico M, Martello S (2009) Assignment problems. In: SIAM monographs on discrete mathematics and applications. Society for Industrial and Applied Mathematics (SIAM), PhiladelphiaGoogle Scholar
  5. Colson B, Marcotte P, Savard G (2005) Bilevel programming: a survey. 4OR: A Q J Oper Res 3: 87–107CrossRefGoogle Scholar
  6. Dempe S (2003) Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52: 333–359CrossRefGoogle Scholar
  7. Deng X (1998) Complexity issues in bilevel linear programming. In: Migdalas A, Pardalos PM, Värband P (eds) Multilevel optimization. Algorithms and applications. Kluwer, Dordrecht, pp 149–164Google Scholar
  8. Deng X (1998) Complexity issues in bilevel linear programming. In: Migdalas A, Pardalos PM, Värband P (eds) Multilevel optimization. Algorithms and applications. Kluwer, Dordrecht, pp 149–164Google Scholar
  9. Garey M, Johnson D (1979) Computers and intractability, a guide to the theory of NP-completeness. Freeman, New YorkGoogle Scholar
  10. Gassner E (2002) Maximal spannende Baumprobleme mit einer Hierarchie von zwei Entscheidungsträgern (in German). Diploma thesis, Department of Mathematics B, University of Technology, Graz, AustriaGoogle Scholar
  11. Hansen P, Jaumard B, Savard G (1992) New branch-and-bound rules for linear bilevel programming. SIAM J Sci Stat Comput 13: 1194–1217CrossRefGoogle Scholar
  12. Jeroslow R (1985) The polynomial hierarchy and a simple model for competitive analysis. Math Program 32: 146–164CrossRefGoogle Scholar
  13. Migdalas A, Pardalos PM, Värband P (1998) Multilevel optimization. Algorithms and applications. Kluwer, DordrechtGoogle Scholar
  14. Stackelberg H von (1934) Marktform und Gleichgewicht. Springer, Berlin [engl. transl.: The theory of market economy. Oxford University Press, New York (1952)]Google Scholar
  15. Vicente LN, Calamai PH (1994) Bilevel and multilevel programming: a bibliography review. J Global Optim 5: 291–306CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Institut für Optimierung und Diskrete Mathematik, TU GrazGrazAustria

Personalised recommendations