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The Distribution of Values in the Quadratic Assignment Problem

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Integer Programming and Combinatorial Optimization (IPCO 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2337))

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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

This research was partially supported by NSF grant DMS 9734138

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1109

    Alexander Barvinok & Tamon Stephen

Authors
  1. Alexander Barvinok
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  2. Tamon Stephen
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Editors and Affiliations

  1. Program in Applied and Computational Mathematics, University of Princeton, Fine Hall, Washington Road, Princeton, NJ, 08544-1000, USA

    William J. Cook

  2. Sloan School of Management, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA

    Andreas S. Schulz

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© 2002 Springer-Verlag Berlin Heidelberg

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Barvinok, A., Stephen, T. (2002). The Distribution of Values in the Quadratic Assignment Problem. In: Cook, W.J., Schulz, A.S. (eds) Integer Programming and Combinatorial Optimization. IPCO 2002. Lecture Notes in Computer Science, vol 2337. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47867-1_26

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  • DOI: https://doi.org/10.1007/3-540-47867-1_26

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43676-8

  • Online ISBN: 978-3-540-47867-6

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