Skip to main content
Springer Nature Link
Log in

Combinatorial Optimization for Weighing Matrices with the Ordering Messy Genetic Algorithm

  • Conference paper
Experimental Algorithms (SEA 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6630))

Included in the following conference series:

  • 2719 Accesses

Abstract

In this paper, we demonstrate that the search for weighing matrices constructed from two circulants can be viewed as a permutation problem. To solve it a set of two competent genetic algorithms (CGAs) are used to locate common integers in two sorted arrays. The motivation to deal with the messy genetic algorithm (mGA) is given from the pioneering results of Goldberg, regarding the ability of the mGA to put tight genes together in a solution which points directly to structural patterns in weighing matrices. In order to take into advantage a recent formalism on the support of two sequences with zero autocorrelation we use an adaptation of the ordering messy GA (OmeGA) where we combine the fast mGA with random keys to represent permutations of the two sequences under investigation. This transformation of the weighing matrices problem to an instance of a combinatorial optimization problem seems to be promising since we illustrate that our framework is capable to solve open cases for weighing matrices as these are listed in the second edition of the Handbook of Combinatorial Designs.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Bean, J.C.: Genetic algorithms and random keys for sequencing and optimization. ORSA J. on Computing 6, 154–160 (1994)

    Article  MATH  Google Scholar 

  2. van Dam, W.: Quantum algorithms for weighing matrices and quadratic residues. Algorithmica 34, 413–428 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Fenimore, E., Cannon, T.: Coded aperture imaging with uniformly redundant array. Appl. Optics 17, 337–347 (1978)

    Article  Google Scholar 

  4. Geramita, A.V., Seberry, J.: Orthogonal Designs. Quadratic Forms and Hadamard Matrices. Lecture Notes in Pure and Applied Mathematics, vol. 45. Marcel Dekker, Inc., New York (1979)

    MATH  Google Scholar 

  5. Golomb, S., Taylor, H.: Two-dimensional synchronization patterns for minimum ambiguity. IEEE Trans. Inform. Theory 28, 600–604 (1982)

    Article  MathSciNet  Google Scholar 

  6. Goldberg, D.E., Deb, K., Korb, B.: Messy genetic algorithms: Motivation, analysis, and first results. Complex Systems 5, 493–530 (1989)

    MathSciNet  MATH  Google Scholar 

  7. Goldberg, D.E., Deb, K., Korb, B.: Messy genetic algorithms revisited: Studies in mixed size and scale. Complex Systems 4, 415–444 (1990)

    MATH  Google Scholar 

  8. Goldberg, D.E., Deb, K., Kargupta, H., Harik, G.: Rapid, Accurate Optimization of Difficult Problems Using Fast Messy Genetic Algorithms. In: Proceedings of the Fifth International Conference on Genetic Algorithms, pp. 56–64. Morgan Kaufmann Publishers Inc., San Francisco (1993)

    Google Scholar 

  9. Goncalves, J.F., Resende, M.G.C.: Biased random-key genetic algorithms for combinatorial optimization (to appear in Journal of Heuristics)

    Google Scholar 

  10. Hersheya, J., Yarlagadda, R.: Two-dimensional synchronisation. Electron. Lett. 19, 801–803 (1983)

    Article  Google Scholar 

  11. Holland, J.H.: Adaptation in Natural and Artificial Systems. An Introductory Analysis with Applications to Biology, Control and Artificial Intelligence. University of Michigan Press, Ann Arbor (1975)

    MATH  Google Scholar 

  12. De Jong, K.A.: An Analysis of the Behavior of a Class of Genetic Adaptive Systems. Doctoral Thesis, CCS Department. University of Michigan, Ann Arbor, MI (1975)

    Google Scholar 

  13. Kargupta, H., Deb, K., Goldberg, D.E.: Ordering Genetic Algorithms and Deception. In: Männer, R., Manderick, B. (eds.) Parallel Problem Solving from Nature PPSN II, pp. 47–56. Elsevier Science Publishers B.V. (1992)

    Google Scholar 

  14. Knjazew, D.: OmeGA: A Competent Genetic Algorithm for Solving Permutation and Scheduling Problems. Kluwer, Norwell (2002)

    Book  MATH  Google Scholar 

  15. Kotsireas, I.S., Koukouvinos, C., Seberry, J.: Weighing matrices and string sorting. Annals of Combinatorics 13, 305–313 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kotsireas, I.S., Koukouvinos, C., Pardalos, P.M.: An efficient string sorting algorithm for weighing matrices of small weight. Optimization Letters 4, 29–36 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kotsireas, I.S., Koukouvinos, C., Pardalos, P.M.: A modified power spectral density test applied to weighing matrices with small weight (to appear in J. Comb. Optim.)

    Google Scholar 

  18. Kotsireas, I.S., Koukouvinos, C., Pardalos, P.M., Simos, D.E.: Competent genetic algorithms for weighing matrices (submitted for publication)

    Google Scholar 

  19. Kotsireas, I.S., Koukouvinos, C., Pardalos, P.M., Shylo, O.: Periodic complementary binary sequences and combinatorial optimization algorithms. J. Comb. Optim. 20, 63–75 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kharaghani, H., Koukouvinos, C.: Complementary, Base and Turyn Sequences. In: Colbourn, C.J., Dinitz, J.H. (eds.) Handbook of Combinatorial Designs, 2nd edn., pp. 317–321. Chapman and Hall/CRC Press, Boca Raton, Fla (2006)

    Google Scholar 

  21. Knuth, D.E.: The Art of Computer Programming, 3rd edn. Seminumerical Algorithms of Addison- Wesley Series in Computer Science and Information Processing, vol. 2. Addison-Wesley Publishing Co., Mass. (1998)

    MATH  Google Scholar 

  22. Koukouvinos, C.: Sequences with Zero Autocorrelation. In: Colbourn, C.J., Dinitz, J.H. (eds.) The CRC Handbook of Combinatorial Designs, pp. 452–456. CRC Press, Boca Raton (1996)

    Google Scholar 

  23. Koukouvinos, C., Seberry, J.: Weighing matrices and their applications. J. Statist. Plann. Inference 62, 91–101 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  24. Koukouvinos, C., Seberry, J.: New weighing matrices and orthogonal designs constructed using two sequences with zero autocorrelation function - a review. J. Statist. Plann. Inference 81, 153–182 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Koukouvinos, C., Simos, D.E.: On the computation of the non-periodic autocorrelation function of two ternary sequences and its related complexity analysis (to appear in J. Appl. Math. & Informatics)

    Google Scholar 

  26. Pardalos, P.M., Du, D.-Z. (eds.): Handbook of Combinatorial Optimization. Combinatorial Optimization, vol. 2. Kluwer Academic Publishers, Springer Netherlands (1998)

    Google Scholar 

  27. Pardalos, P.M., Resende, M.G.C. (eds.): Handbook of Applied Optimization. Oxford University Press, Inc., 198 Madison Avenue, USA (2002)

    MATH  Google Scholar 

  28. Rothlauf, F.: Representations for Genetic and Evolutionary Algorithms, 2nd edn. Physica-Verlag, Heidelberg (2006)

    MATH  Google Scholar 

  29. Seberry, J., Yamada, M.: Hadamard Matrices, Sequences and Block Designs. In: Dinitz, J.H., Stinson, D.R. (eds.) Contemporary Design Theory: A Collection of Surveys, pp. 431–560. John Wiley & Sons, New York (1992)

    Google Scholar 

  30. Weathers, G., Holiday, E.M.: Group-complementary array coding for radar clutter rejection. IEEE Transaction on Aerospace and Electronic Systems 19, 369–379 (1983)

    Article  Google Scholar 

  31. Wallis, J.S.: On supplementary difference sets. Aequationes Math. 8, 242–257 (1972)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Zografou, 15773, Athens, Greece

    Christos Koukouvinos & Dimitris E. Simos

Authors
  1. Christos Koukouvinos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dimitris E. Simos
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA

    Panos M. Pardalos

  2. Division of Economics and Business, Colorado School of Mines, Engineering Hall 310, 816 15th Street, 80401, Golden, CO, USA

    Steffen Rebennack

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Koukouvinos, C., Simos, D.E. (2011). Combinatorial Optimization for Weighing Matrices with the Ordering Messy Genetic Algorithm. In: Pardalos, P.M., Rebennack, S. (eds) Experimental Algorithms. SEA 2011. Lecture Notes in Computer Science, vol 6630. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20662-7_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-20662-7_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-20661-0

  • Online ISBN: 978-3-642-20662-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics