The Distribution of Values in the Quadratic Assignment Problem

  • Alexander Barvinok
  • Tamon Stephen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2337)


We obtain a number of results regarding the distribution of values of a quadratic function f on the set of n x n permutation matrices (identified with the symmetric group Sn S n ) around its optimum (minimum or maximum). We estimate the fraction of permutations σ such that f(σ) lies within a given neighborhood of the optimal value of f and relate the optimal value with the average value of f over a neighborhood of the optimal permutation. We describe a natural class of functions (which includes, for example, the objective function in the Traveling Salesman Problem) with a relative abundance of near-optimal permutations. Also, we identify a large class of functions f with the property that permutations close to the optimal permutation in the Hamming metric of S n tend to produce near optimal values of f, and show that for general f just the opposite behavior may take place


Symmetric Group Travel Salesman Problem Travel Salesman Problem Random Permutation Extreme Function 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anstreicher, K., Brixius, N., Goux, J.-P., Linderoth, J.: Solving large quadratic assignment problems on computational grids. Math Programming B (to appear)Google Scholar
  2. 2.
    Arkin, E., Hassin, R., Sviridenko, M.: Approximating the maximum quadratic assignment problem. Inform. Process. Lett. 77 (2000) no. 1. 13–16MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and approximation. Combinatorial optimization problems and their approximability properties. Springer-Verlag, Berlin (1999)Google Scholar
  4. 4.
    Brüngger, A., Marzetta, A., Clausen, J., Perregaard M.: Solving large scale quadratic assignment problems in parallel with the search library ZRAM. Journal of Parallel and Distributed Computing 50 (1998) 157–66zbMATHCrossRefGoogle Scholar
  5. 5.
    Burkard, R., çCela, E., Pardalos, P., Pitsoulis, L.: The quadratic assignment problem. In: Du, D.-Z., and Pardalos, P. M. (eds.): Handbook of Combinatorial Optimization. Kluwer Academic Publishers (1999) 75–149Google Scholar
  6. 6.
    Fulton, W., Harris, J.: Representation theory. Springer-Verlag, New York (1991)zbMATHCrossRefGoogle Scholar
  7. 7.
    Goulden, I. P., Jackson, D. M.: Combinatorial enumeration. John Wiley & Sons, Inc., New York (1983)zbMATHGoogle Scholar
  8. 8.
    Graves, G. W., Whinston, A. B.: An algorithm for the quadratic assignment problem. Management Science 17 (1970) no. 7. 452–71Google Scholar
  9. 9.
    Stephen, T.: The distribution of values in combinatorial optimization problems. Ph.D. Dissertation, University of Michigan (in preparation)Google Scholar
  10. 10.
    Ye, Y.: Approximating quadratic programming with bound and quadratic constraints. Math. Programming 84 (1999) no. 2. 219–226zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Alexander Barvinok
    • 1
  • Tamon Stephen
    • 1
  1. 1.Department of MathematicsUniversity of MichiganAnn Arbor

Personalised recommendations