Mathematical Programming

, Volume 82, Issue 1–2, pp 125–158 | Cite as

The quadratic assignment problem with a monotone anti-Monge and a symmetric Toeplitz matrix: Easy and hard cases

  • Rainer E. Burkard
  • Eranda Çela
  • Günter Rote
  • Gerhard J. Woeginger
Article

Abstract

This paper investigates a restricted version of the Quadratic Assignment Problem (QAP), where one of the coefficient matrices is an Anti-Monge matrix with non-decreasing rows and columns and the other coefficient matrix is a symmetric Toeplitz matrix. This restricted version is called the Anti-Monge—Toeplitz QAP. There are three well-known combinatorial problems that can be modeled via the Anti-Monge—Toeplitz QAP: (Pl) The “Turbine Problem”, i.e. the assignment of given masses to the vertices of a regular polygon such that the distance of the center of gravity of the resulting system to the center of the polygon is minimized. (P2) The Traveling Salesman Problem on symmetric Monge distance matrices. (P3) The arrangement of data records with given access probabilities in a linear storage medium in order to minimize the average access time. We identify conditions on the Toeplitz matrixB that lead to a simple solution for the Anti-Monge—Toeplitz QAP: The optimal permutation can be given in advance without regarding the numerical values of the data. The resulting theorems generalize and unify several known results on problems (P1), (P2), and (P3). We also show that the Turbine Problem is NP-hard and consequently, that the Anti-Monge—Toeplitz QAP is NP-hard in general. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Keywords

Quadratic assignment Special cases Polynomially solvable Anti-Monge matrices Toeplitz matrices 

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Copyright information

© The Mathematical Programming Society, Inc. 1998

Authors and Affiliations

  • Rainer E. Burkard
    • 1
  • Eranda Çela
    • 1
  • Günter Rote
    • 1
  • Gerhard J. Woeginger
    • 1
  1. 1.Institut für Mathematik BTechnische Universität GrazGrazAustria

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