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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)

Abstract

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

Keywords

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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

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

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