Mathematical Programming

, Volume 82, Issue 1–2, pp 125–158

# 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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### References

1. [1]
S.V. Amiouny, J.J. Bartholdi III, J.H. Vande Vate, J. Zhang, Balanced loading, Operations Research 40 (1992) 238–246.Google Scholar
2. [2]
A.A. Bolotnikov, On the best balance of the disk with masses on its periphery, Problemi Mashinostroenia 6 (1978) 68–74 (in Russian).Google Scholar
3. [3]
R.E. Burkard, Locations with spatial interactions: the quadratic assignment problem, in: P.B. Mirchandani, R.L. Francis (Eds.), Discrete Location Theory, ch. 9, Wiley, New York, 1990, pp. 387–437.Google Scholar
4. [4]
R.E. Burkard, E. Çela, G. Rote, G.J. Woeginger, The quadratic assignment problem with an Anti-Monge and a Toeplitz matrix: easy and hard cases, Technical Report SFB-34, June 1995, 30 pages, Institut für Mathematik B, Technische Universität Graz. file://ftp.tu-graz.ac.at/pub/papers/math/sfb34.ps.gz.Google Scholar
5. [5]
R.E. Burkard, B. Klinz, R. Rudolf, Perspectives of Monge properties in optimization, Discrete Applied Mathematics 70 (1996) 95–161.Google Scholar
6. [6]
V.N. Burkov, M.I. Rubinstein, V.B. Sokolov, Some problems in optimal allocation of large-volume memories, Avtomatika i Telemekhanika 9 (1969) 83–91 (in Russian).Google Scholar
7. [7]
E. Çela, G.J. Woeginger, A note on the maximum of a certain bilinear form, Technical Report SFB-8, September 1994, Institut für Mathematik B, Technische Universität Graz. file://ftp.tu-graz.ac.at/pub/papers/math/sfb8.ps.gz.Google Scholar
8. [8]
J. Clausen, M. Perregård, Solving large quadratic assignment problems in parallel, Computational Optimization and Applications 8 (1997) 111–127.Google Scholar
9. [9]
M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco, 1979.Google Scholar
10. [10]
G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1967.Google Scholar
11. [11]
G.H. Hardy, J.E. Littlewood, G. Pólya, The maximum of a certain bilinear form, Proceedings of the London Mathematical Society 25 (1926) 265–282.Google Scholar
12. [12]
T.C. Koopmans, M.J. Beckmann, Assignment problems and the location of economic activities, Econometrica 25 (1957) 53–76.Google Scholar
13. [13]
G. Laporte, H. Mercure, Balancing hydraulic turbine runners: a quadratic assignment problem, European Journal of Operational Research 35 (1988) 378–382.Google Scholar
14. [14]
E.L. Lawler, The quadratic assignment problem, Management Science 9 (1963) 586–599.Google Scholar
15. [15]
E.L. Lawler, The quadratic assignment problem: a brief review, in: B. Roy (Ed.), Combinatorial Programming: Methods and Applications, Reidel Dordrecht, Holland, 1975, pp. 351–360.Google Scholar
16. [16]
E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, D.B. Shmoys, The Traveling Salesman Problem, Wiley, Chichester, 1985.Google Scholar
17. [17]
N.N. Metelski, On extremal values of quadratic forms on symmetric groups, Vesti Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk 6 (1972) 107–110 (in Russian).Google Scholar
18. [18]
J. Mosevich, Balancing hydraulic turbine runners — a discrete combinatorial optimization problem, European Journal of Operational Research 26 (1986) 202–204.Google Scholar
19. [19]
P. Pardalos, F. Rendl, H. Wolkowicz, The quadratic assignment problem: a survey and recent developments, in: P. Pardalos, H. Wolkowicz (Eds.), Proceedings of the DIMACS Workshop on Quadratic Assignment Problems, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 16 (1994) 1–42.Google Scholar
20. [20]
V.R. Pratt, AnN logN algorithm to distributeN records optimally in a sequential access file, in: R.E. Miller, J.W. Thatcher (Eds.), Complexity of Computer Computations, Plenum Press, New York, 1972, pp. 111–118.Google Scholar
21. [21]
D. Schlegel, Die Unwucht-optimale Verteilung von Turbinenschaufeln als quadratisches Zuordnungsproblem, Ph.D. Thesis, ETH Zürich, 1987.Google Scholar
22. [22]
Y.G. Stoyan, V.Z. Sokolovskii, S.V. Yakovlev, A method for balancing discretely distributed masses under rotation, Energomashinostroenia 2 (1982) 4–5 (in Russian).Google Scholar
23. [23]
F. Supnick, Extreme Hamiltonian lines, Annals of Mathematics 66 (1957) 179–201.Google Scholar
24. [24]
B.B. Timofeev, V.A. Litvinov, On the extremal value of a quadratic form, Kibernetika 4 (1969) 56–61 (in Russian).Google Scholar

© 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