Advertisement

Polyhedral Methods for Solving Three Index Assignment Problems

  • Liqun Qi
  • Defeng Sun
Part of the Combinatorial Optimization book series (COOP, volume 7)

Abstract

The (axial) three index assignment problem, also known as the three-dimensional matching problem, is the problem of assigning one item to one job at one point or interval of time in such a way as to minimize the total cost of the assignment. Until now the most efficient algorithms explored for solving this problem are based on polyhedral combinatorics. So far, four important facet classes Q, P, B and C have been characterized and O(n3 )(linear-time) separation algorithms for five facet subclasses of Q, P and B have been established. The complexity of these separation algorithms is best possible since the number of the variables of three index assignment problem of order n is n3. In this paper, we review these progresses and raise some further questions on this topic.

Keywords

Assignment Problem Linear Programming Relaxation Separation Algorithm Facet Class Multidimensional Assignment Problem 
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. [Balas and Qi, 1993]
    Balas, E. and Qi, L. (1993). Linear-time separation algorithms for the three-index assignment polytope. Discrete Applied Mathematics, 43:1–12.MathSciNetMATHCrossRefGoogle Scholar
  2. [Balas and Saltzman, 1989]
    Balas, E. and Saltzman, M. (1989). Facets of the three-index assignment polytope. Discrete Applied Mathematics, 23:201–229.MathSciNetMATHCrossRefGoogle Scholar
  3. [Balas and Saltzman, 1991]
    Balas, E. and Saltzman, M. (1991). An algorithm for the three-index assignment problem. Oper. Res., 39:150–161.MathSciNetMATHCrossRefGoogle Scholar
  4. [Burkard and Fröhlich, 1980]
    Burkard, R. and Fröhlich, K. (1980). Some remarks on the three-dimensional assignment problem. Method of Oper. Res., 36:31–36.MATHGoogle Scholar
  5. [Burkard and Rudolf, 1997]
    Burkard, R. and Rudolf, R. (1997). Computational investigations on 3-dimensional axial assignment problems. Belgian Journal of Operations Research, Statistics and Computer Science.Google Scholar
  6. [Euler, 1987]
    Euler, R. (1987). Odd cycles and a class of facets of the axial 3-index assignment polytope. Applicationes Mathematicae (Zastosowania Matematyki), XIX:375–386.Google Scholar
  7. [Frieze, 1983]
    Frieze, A. (1983). Complexity of a 3-dimensional assignment problem. European J. Oper. Res., 13:161–164.MathSciNetMATHCrossRefGoogle Scholar
  8. [Fröhlich, 1979]
    Fröhlich, K. (1979). Dreidimenionale Zuordnungsprobleme. Diplomarbeit, Mathematisches Institut, Universität zu Köln, Germany.Google Scholar
  9. [Garey and Johnson, 1979]
    Garey, M. and Johnson, D. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco, CA.MATHGoogle Scholar
  10. [Gwan, 1993]
    Gwan, G. (1993). A Polyhedral Method for the Three Index Assignment Problem. PhD thesis, School of Mathematics, University of New South Wales, Sydney 2051, Australia.Google Scholar
  11. [Gwan and Qi, 1992]
    Gwan, G. and Qi, L. (1992). On facets of the three-index assignment polytope. Australasian J. Combinatorics, 6:67–87.MathSciNetMATHGoogle Scholar
  12. [Hansen and Kaufman, 1973]
    Hansen, P. and Kaufman, L. (1973). A primaldual algorithm for the three-dimensional assignment problem. Cahiers du CERO, 15:327–336.MathSciNetMATHGoogle Scholar
  13. [Qi et al., 1994]
    Qi, L., Balas, E. and Gwan, G. (1994). A new facet class and a polyhedral method for the three-index assignment problem. In Advances in Optimization and Approximation, pages 256–274. Kluwer Academic Publishers, Dordrecht.CrossRefGoogle Scholar
  14. [Leue, 1972]
    Leue, O. (1972). Methoden zur lösung dreidimensionaler zuordnungsprobleme. Angewandte Informatik, pages 154–162.Google Scholar
  15. [Nemhauser and Wolsey, 1988]
    Nemhauser, G. L. and Wolsey, L. A. (1988). Integer and Combinatorial Optimization. John Wiley & Sons, New York.MATHGoogle Scholar
  16. [Pierskalla, 1967]
    Pierskalla, W. (1967). The tri-substitution method for the three-dimensional assignment problem. CORS J., 5:71–81.Google Scholar
  17. [Pierskalla, 1968]
    Pierskalla, W. (1968). The multidimensional assignment problem. Oper. Res., 16:422–431.MATHCrossRefGoogle Scholar
  18. [Burkard et al., 1996]
    Burkard, R.E., Rüdiger, R., and Gerhard, W. (1996). Three-dimensional axial assignment problems with decomposable cost coefficients. Discrete Appl. Math., 65:123–139.MathSciNetMATHCrossRefGoogle Scholar
  19. [Rudolf, 1991]
    Rudolf, R. (1991). Dreidimensionale axiale Zuordnungsprobleme. Technische Universität Graz, Austria. Master Thesis.Google Scholar
  20. [Vlach, 1967]
    Vlach, M. (1967). Branch and bound method for the three index assignment problem. Ekonomicko-Mathematický Obzor, 3:181–191.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Liqun Qi
    • 1
  • Defeng Sun
    • 1
  1. 1.School Of MathematicsThe University of New South WalesSydneyAustralia

Personalised recommendations