Optimization Letters

, Volume 10, Issue 6, pp 1303–1314 | Cite as

Circulant weighing matrices: a demanding challenge for parallel optimization metaheuristics

  • D. Souravlias
  • K. E. Parsopoulos
  • I. S. Kotsireas
Original Paper


Circulant weighing matrices constitute a special type of combinatorial matrices that have attracted scientific interest for many years. The existence and determination of specific classes of circulant weighing matrices remains an active research area that involves both theoretical algebraic techniques as well as high-performance computational optimization approaches. The present work aims at investigating the potential of four established parallel metaheuristics as well as a special Algorithm Portfolio approach, on solving such problems. For this purpose, the algorithms are applied on a hard circulant weighing matrix existence problem. The obtained results are promising, offering insightful conclusions.


Circulant weighing matrices Parallel metaheuristics  Algorithm portfolios 



The authors would like to thank the Shared Hierarchical Academic Research Computing Network (SHARCNET) as well as the WestGrid HPC consortium for offering the necessary computational resources.


  1. 1.
    Alba, E.: Parallel Metaheuristics: A New Class of Algorithms. Wiley, London (2005)CrossRefzbMATHGoogle Scholar
  2. 2.
    Ang, M., Arasu, K., Ma, S., Strassler, Y.: Study of proper circulant weighing matrices with weigh 9. Discrete Math. 308, 2802–2809 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arasu, K., Dillon, J., Jungnickel, D., Pott, A.: The solution of the waterloo problem. J. Comb. Theory Ser. A 71, 316–331 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Arasu, K., Gulliver, T.: Self-dual codes over fp and weighing matrices. IEEE Trans. Inf. Theory 47(5), 2051–2055 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Arasu, K., Gutman, A.: Circulant weighing matrices. Cryptogr. Commun. 2, 155–171 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Arasu, K., Leung, K., Ma, S., Nabavi, A., Ray-Chaudhuri, D.: Determination of all possible orders of weight 16 circulant weighing matrices. Finite Fields Appl. 12, 498–538 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chiarandini, M., Kotsireas, I., Koukouvinos, C., Paquete, L.: Heuristic algorithms for hadamard matrices with two circulant cores. Theor. Comput. Sci. 407(1–3), 274–277 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cousineau, J., Kotsireas, I., Koukouvinos, C.: Genetic algorithms for orthogonal designs. Australas. J. Comb. 35, 263–272 (2006)MathSciNetzbMATHGoogle Scholar
  9. 9.
    van Dam, W.: Quantum algorithms for weighing matrices and quadratic residues. Algorithmica 34, 413–428 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Eades, P.: On the existence of orthogonal designs. Ph.D. thesis, Australian National University, Canberra (1997)Google Scholar
  11. 11.
    Eades, P., Hain, R.: On circulant weighing matrices. Ars Comb. 2, 265–284 (1976)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Eberhart, R.C., Kennedy, J.: A new optimizer using particle swarm theory. In: Proceedings Sixth Symposium on Micro Machine and Human Science, pp. 39–43. Piscataway, NJ (1995)Google Scholar
  13. 13.
    Geramita, A., Sebery, J.: Orthogonical designs: quadratic forms and hadamard matrices. Lecture Notes in Pure and Applied Mathematics (1979)Google Scholar
  14. 14.
    Glover, F.: Future paths for integer programming and links to artificial intelligence. Comput. Oper. Res. 13(5), 533–549 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gomes, C.P., Selman, B.: Algorithm portfolio design: theory vs. practice. In: Proceedings Thirteenth conference on Uncertainty in artificial intelligence, pp. 190–197 (1997)Google Scholar
  16. 16.
    Hansen, P., Mladenović, N., Brimberg, J., Moreno Pérez, J.A.: Variable neighborhood search. In: M. Gendreau, J.Y. Potvin (eds.) Handbook of Metaheuristics, vol. 146, chap. 3. Springer, Berlin (2010)Google Scholar
  17. 17.
    Huberman, B.A., Lukose, R.M., Hogg, T.: An economics approach to hard computational problems. Science 27, 51–53 (1997)CrossRefGoogle Scholar
  18. 18.
    Kotsireas, I.: Algorithms and metaheuristics for combinatorial matrices. In: P. Pardalos, D.Z. Du, R.L. Graham (eds.) Handbook of Combinatorial Optimization, pp. 283–309. Springer, New York (2013)Google Scholar
  19. 19.
    Kotsireas, I., Koukouvinos, C., Pardalos, P., Shylo, O.: Periodic complementary binary sequences and combinatorial optimization algorithms. J. Combin. Optim. 20(1), 63–75 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kotsireas, I., Koukouvinos, C., Pardalos, P., Simos, D.: Competent genetic algorithms for weighing matrices. J. Combin. Optim. 24(4), 508–525 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kotsireas, I., Parsopoulos, K., Piperagkas, G., Vrahatis, M.: Ant-based approaches for solving autocorrelation problems. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 7461 LNCS, pp. 220–227 (2012)Google Scholar
  22. 22.
    Koukouvinos, C., Seberry, J.: Weighing matrices and their applications. J. Stat. Plan. Inference 62(1), 91–101 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mladenovic, N., Hansen, P.: Variable neighborhood search. Comput. Oper. Res. 24(11), 1097–1100 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Parsopoulos, K.E., Vrahatis, M.N.: Particle Swarm Optimization and Intelligence: Advances and Applications. Information Science Publishing (IGI Global), Hershey, USA (2010)Google Scholar
  25. 25.
    Peng, F., Tang, K., Chen, G., Yao, X.: Population-based algorithm portfolios for numerical optimization. IEEE Trans. Evol. Comput. 14(5), 782–800 (2010)CrossRefGoogle Scholar
  26. 26.
    Ribeiro, C., Resende, M.: Path-relinking intensification methods for stochastic local search algorithms. J. Heuristics 18(2), 193–214 (2012)CrossRefGoogle Scholar
  27. 27.
    Schmidt, B., Smith, K.W.: Circulant weighing matrices whose order and weight are products of powers of 2 and 3. J. Comb. Theory Ser. A 120(1), 275–287 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Seberry, J., Whiteman, A.: Some results on weighing matrices. Bull. Aust. Math. Soc. 12, 433–447 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Souravlias, D., Parsopoulos, K.E., Alba, E.: Parallel algorithm portfolio with market trading-based time allocation. In: Proceedings International Conference on Operations Research 2014 (OR2014) (2014)Google Scholar
  30. 30.
    Storn, R., Price, K.: Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 11(4), 341–359 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Strassler, Y.: The classification of circulant weighing matrices of weight 9. Ph.D. thesis, Bar-Ilan University (1997)Google Scholar
  32. 32.
    Tang, K., Peng, F., Chen, G., Yao, X.: Population-based algorithm portfolios with automated constituent algorithms selection. Inf. Sci. 279, 94–104 (2014)CrossRefGoogle Scholar
  33. 33.
    Tasgetiren, F., Chen, A., Gencyilmaz, G., Gattoufi, S.: Smallest position value approach. Stud. Comput. Intel. 175, 121–138 (2009)zbMATHGoogle Scholar
  34. 34.
    Yevseyeva, I., Guerreiro, A.P., Emmerich, M.T.M., Fonseca, C.M.: A portfolio optimization approach to selection in multiobjective evolutionary algorithms. Proc. PPSN 2014, 672–681 (2014)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • D. Souravlias
    • 1
  • K. E. Parsopoulos
    • 1
  • I. S. Kotsireas
    • 2
  1. 1.Department of Computer Science and EngineeringUniversity of IoanninaIoanninaGreece
  2. 2.Department of Physics and Computer ScienceWilfrid Laurier UniversityWaterlooCanada

Personalised recommendations