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Polyhedral Methods for the QAP

  • Volker Kaibel
Part of the Combinatorial Optimization book series (COOP, volume 7)

Abstract

For many combinatorial optimization problems investigations of associated polyhedra have led to enormous successes with respect to both theoretical insights into the structures of the problems as well as to their algorithmic solvability. Among these problems are quite prominent NP-hard ones, like, e.g., the traveling salesman problem, the stable set problem, or the maximum cut problem. In this chapter we overview the polyhedral work that has been done on the quadratic assignment problem (QAP). Our treatment includes a brief introduction to the methods of polyhedral combinatorics in general, descriptions of the most important polyhedral results that are known about the QAP, explanations of the techniques that are used to prove such results, and a discussion of the practical results obtained by cutting plane algorithms that exploit the polyhedral knowledge. We close by some remarks on the perspectives of this kind of approach to the QAP.

Keywords

Assignment Problem Combinatorial Optimization Problem Quadratic Assignment Problem Incidence Vector Linear Description 
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.

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References

  1. [Adams and Johnson, 1994]
    Adams, W. P. and Johnson, T. A. (1994). Improved linear programming-based lower bounds for the quadratic assignment problem. In Pardalos, P. M. and Wolkowicz, H., editors, Quadratic Assignment and Related Problems, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 43–75.Google Scholar
  2. [Barvinok, 1992]
    Barvinok, A. I. (1992). Combinatorial complexity of orbits in representations of the symmetric group. Advances in Soviet Mathematics, 9:161–182.MathSciNetGoogle Scholar
  3. [Brüngger et al., 1996]
    Brüngger, A., Clausen, J., Marzetta, A., and Perregaard, M. (1996). Joining forces in solving large-scale quadratic assignment problems in parallel. Technical Report DIKU TR-23/96, University of Copenhagen, Copenhagen.Google Scholar
  4. [Burkard et al., 1997]
    Burkard, R. E., Karisch, S. E., and Rendl, F. (1997). QAPLIB A quadratic assignment problem library. Journal of Global Optimization, 10:391–403. http://serv1.imm.dtu.dk/~sk/qaplib/. MathSciNetMATHCrossRefGoogle Scholar
  5. [Chvátal, 1983]
    Chvátal, V. (1983). Linear Programming. Freeman.Google Scholar
  6. [Clausen and Perregaard, 1997]
    Clausen, J. and Perregaard, M. (1997). Solving large scale quadratic assignment problems in parallel. Comput. Optim. Appl., 8(2):111–127.MathSciNetMATHCrossRefGoogle Scholar
  7. [Deza and Laurent, 1997]
    Deza, M. M. and Laurent, M. (1997). Geometry of Cuts and Metrics. Springer Verlag.MATHGoogle Scholar
  8. [Edmonds, 1965a]
    Edmonds, J. (1965a). Maximum matching and a polyhedron with 0,1-vertices. Journal of Research of the National Bureau of Standards—B, Mathematics and Mathematical Physics, 69B:125–130.MathSciNetGoogle Scholar
  9. [Edmonds, 1965b]
    Edmonds, J. (1965b). Paths, trees, and flowers. Canadian Journal of Mathematics, 17:449–467.MathSciNetMATHCrossRefGoogle Scholar
  10. [Elf, 1999]
    Elf, M. (1999). LP-basierte Schranken für quadratische Zuordnungsprobleme mit dünner Zielfunktion. Master’s thesis, Universität zu Köln.Google Scholar
  11. [Gilmore, 1962]
    Gilmore, P. C. (1962). Optimal and suboptimal algorithms for the quadratic assignment problem. SIAM Journal on Applied Mathematics, 10:305–313.MathSciNetMATHCrossRefGoogle Scholar
  12. [Grötschel et al., 1981]
    Grötschel, M., Lovász, L., and Schrijver, A. (1981). The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1:169–197.MathSciNetMATHCrossRefGoogle Scholar
  13. [Grötschel et al., 1988]
    Grötschel, M., Lovász, L., and Schrijver, A. (1988). Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, Heidelberg.MATHCrossRefGoogle Scholar
  14. [Grünbaum, 1967]
    Grünbaum, B. (1967). Convex Polytopes. Interscience, London. Revised edition: V. Klee and P. Kleinschmidt, Graduate Texts in Mathematics, Springer Verlag.MATHGoogle Scholar
  15. [Johnson, 1992]
    Johnson, T. A. (1992). New Linear-Programming Based Solution Procedures for the Quadratic Assignment Problem. PhD thesis, Graduate School of Clemson University.Google Scholar
  16. [Jünger and Kaibel, 1996]
    Jünger, M. and Kaibel, V. (1996). On the SQAPpolytope. Technical Report No. 96.241, Institut für Informatik, Universität zu Köln. Submitted to: SIAM Journal on Optimization.Google Scholar
  17. [Jünger and Kaibel, 1997a]
    Jünger, M. and Kaibel, V. (1997a). Boxinequalities for quadratic assignment polytopes. Technical Report 97.285, Angewandte Mathematik und Informatik, Universität zu Köln. Submitted to: Mathematical Programming.Google Scholar
  18. [Jünger and Kaibel, 1997b]
    Jünger, M. and Kaibel, V. (1997b). The QAPpolytope and the star-transformation. Technical Report 97.284, Angewandte Mathematik und Informatik, Universität zu Köln. Submitted to: Discrete Applied Mathematics.Google Scholar
  19. [Jünger et al., 1994]
    Jünger, M., Reinelt, G., and Thienel, S. (1994). Provably good solutions for the traveling salesman problem. ZOR — Mathematical Methods of operations Research, 40:183–217.MATHCrossRefGoogle Scholar
  20. [Kaibel, 1997]
    Kaibel, V. (1997). Polyhedral Combinatorics of the Quadratic Assignment Problem. PhD thesis, Universität zu Köln. wwv.math.TU-Berlin.de/~kaibel/diss.html.Google Scholar
  21. [Kaibel, 1998]
    Kaibel, V. (1998). Polyhedral combinatorics of QAPs with less objects than locations. In Bixby, R. E., Boyd, E. A., and Rios-Mercado, R. Z., editors, Proceedings of the 6th International IPCO Conference, Houston, Texas., volume 1412 of Lecture Notes in Computer Science, pages 409–422. Springer-Verlag.Google Scholar
  22. [Karp and Papadimitriou, 1982]
    Karp, R. M. and Papadimitriou, C. H. (1982). On linear characterizations of combinatorial optimization problems. SIAM Journal on Computing, 11:620–632.MathSciNetMATHCrossRefGoogle Scholar
  23. [Khachiyan, 1979]
    Khachiyan, L. G. (1979). A polynomial algorithm in linear programming. Soviet. Math. Dokl., 20:191–194.MATHGoogle Scholar
  24. [Klee and Kleinschmidt, 1995]
    Klee, V. and Kleinschmidt, P. (1995). Convex polytopes and related complexes. In Graham, R. L., Grötschel, M., and Lovász, L., editors, Handbook of Combinatorics, volume 2, chapter 18, pages 875–918. Elsevier Science.Google Scholar
  25. [Koopmans and Beckmann, 1957]
    Koopmans, T. C. and Beckmann, M. J. (1957). Assignment problems and the location of economic activities. Econometrica, 25:53–76.MathSciNetMATHCrossRefGoogle Scholar
  26. [Lawler, 1963]
    Lawler, E. L. (1963). The quadratic assignment problem. Management Science, 9:586–599.MathSciNetMATHCrossRefGoogle Scholar
  27. [Minkowski, 1896]
    Minkowski, H. (1896). Geometrie der Zahlen. Teubner Verlag, Leipzig. Reprinted by Chelsea, New York 1953 and Johnson, New York 1963.Google Scholar
  28. [Nemhauser and Wolsey, 1988]
    Nemhauser, G. L. and Wolsey, L. A. (1988). Integer and Combinatorial Optimization. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, Chichester New York.MATHGoogle Scholar
  29. [Padberg and Rijal, 1996]
    Padberg, M. and Rijal, M. P. (1996). Location, Scheduling, Design and Integer Programming. Kluwer Academic Publishers.MATHCrossRefGoogle Scholar
  30. [Padberg, 1995]
    Padberg, M. W. (1995). Linear Optimization and Extensions. Springer-Verlag, Berlin Heidelberg.MATHGoogle Scholar
  31. [Padberg and Rao, 1980]
    Padberg, M. W. and Rao, M. R. (1980). The russian method and integer programming. Technical report, CBA Working Paper, New York University, New York.Google Scholar
  32. [Resende et al., 1995]
    Resende, M. G. C., Rarnakrishnan, K. G., and Drezner, Z. (1995). Computing lower bounds for the quadratic assignment problem with an interior point solver for linear programming. Operations Research, 43:781–791.MathSciNetMATHCrossRefGoogle Scholar
  33. [Rijal, 1995]
    Rijal, M. P. (1995). Scheduling, Design and Assignment Problems with Quadratic Costs. PhD thesis, New York University.Google Scholar
  34. [Schrijver, 1986]
    Schrijver, A. (1986). Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons, Chichester New York.MATHGoogle Scholar
  35. [Schrijver, 1995]
    Schrijver, A. (1995). Polyhedral combinatorics. In Graham, R. L., Grötschel, M., and Lovász, L., editors, Handbook of Combinatorics, volume 2, chapter 30, pages 1649–1704. Elsevier Science.Google Scholar
  36. [Weyl, 1935]
    Weyl, H. (1935). Elementare Theorie der konvexen Polyeder. Comm. Math. Helv., 7:290–306.MathSciNetMATHCrossRefGoogle Scholar
  37. [Wolsey, 1998]
    Wolsey, L. A. (1998). Integer Programming. John Wiley &; Sons, Inc.MATHGoogle Scholar
  38. [Ziegler, 1995]
    Ziegler, G. M. (1995). Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag, New York. (Revised edition: 1998).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Volker Kaibel
    • 1
  1. 1.Fachbereich Mathematik, Sekr. 7-1Technische Universität BerlinBerlinGermany

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