Computational Optimization and Applications

, Volume 60, Issue 3, pp 789–814 | Cite as

Addressing the envelope reduction of sparse matrices using a genetic programming system

  • Behrooz KoohestaniEmail author
  • Riccardo Poli


Large sparse symmetric matrix problems arise in a number of scientific and engineering fields such as fluid mechanics, structural engineering, finite element analysis and network analysis. In all such problems, the performance of solvers depends critically on the sum of the row bandwidths of the matrix, a quantity known as envelope size. This can be reduced by appropriately reordering the rows and columns of the matrix, but for an \(N\times N\) matrix, there are \(N!\) such permutations, and it is difficult to predict how each permutation affects the envelope size without actually performing the reordering of rows and columns. These two facts compounded with the large values of \(N\) used in practical applications, make the problem of minimising the envelope size of a matrix an exceptionally hard one. Several methods have been developed to reduce the envelope size. These methods are mainly heuristic in nature and based on graph-theoretic concepts. While metaheuristic approaches are popular alternatives to classical optimisation techniques in a variety of domains, in the case of the envelope reduction problem, there has been a very limited exploration of such methods. In this paper, a Genetic Programming system capable of reducing the envelope size of sparse matrices is presented and evaluated against four of the best-known and broadly used envelope reduction algorithms. The results obtained on a wide-ranging set of standard benchmarks from the Harwell–Boeing sparse matrix collection show that the proposed method compares very favourably with these algorithms.


Genetic programming Envelope reduction problem Sparse matrices Graph labelling Combinatorial optimisation 


  1. 1.
    Pissanetskey, S.: Sparse Matrix Technology. Academic Press, London (1984)Google Scholar
  2. 2.
    Irons, B.M.: A frontal solution program for finite element analysis. Int. J. Numer. Methods. Eng. 2, 5–32 (1970)CrossRefzbMATHGoogle Scholar
  3. 3.
    Jennings, A.: Matrix Computation for Engineers and Scientists. John Wiley, Hoboken (1977)zbMATHGoogle Scholar
  4. 4.
    Barnard, S.T., Pothen, A., Simon, H.: A spectral algorithm for envelope reduction of sparse matrices. Numer. Lin. Algebra. Appl. 2(4), 317–334 (1995)CrossRefzbMATHGoogle Scholar
  5. 5.
    Gary, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. W. H. Freeman and Company, New York (1979)Google Scholar
  6. 6.
    George, A., Pothen, A.: An analysis of spectral envelope reduction via quadratic assignment problems. SIAM J. Matrix Anal. Appl. 18(3), 706–732 (1997)CrossRefzbMATHGoogle Scholar
  7. 7.
    Lin, Y., Yuan, J.: Profile minimization problem for matrices and graphs. Acta Math. Appl. Sin. 10, 107–112 (1994)CrossRefzbMATHGoogle Scholar
  8. 8.
    Koza, J.R.: Genetic Programming: On the Programming of Computers by Means of Natural Selection. MIT Press, Cambridge, MA (1992)zbMATHGoogle Scholar
  9. 9.
    Poli, R., Langdon, W., McPhee, N.: A field guide to genetic programming. Lulu Enterprises, Raleigh (2008).Google Scholar
  10. 10.
    Cuthill, E., McKee, J.: Reducing the bandwidth of sparse symmetric matrices. In: ACM National Conference, pp. 157–172. Association for Computing Machinery, New York (1969).Google Scholar
  11. 11.
    Liu, W.H., Sherman, A.H.: Comparative analysis of the cuthill-mckee and the reverse Cuthill–Mckee ordering algorithms for sparse matrices. SIAM J. Numer. Anal. 13(2), 198–213 (1976)CrossRefzbMATHGoogle Scholar
  12. 12.
    George, J.A.: Computer implementation of the finite element method. Ph.D. thesis, Stanford, CA (1971).Google Scholar
  13. 13.
    Cuthill, E.: Several strategies for reducing the bandwidth of matrices. In: Rose, D., Willoughby, R. (eds.) Sparse Matrices and their Applications. The IBM Research Symposia Series, pp. 157–166. Springer, US (1972)CrossRefGoogle Scholar
  14. 14.
    Gibbs, N.E., Poole, W.G., Stockmeyer, P.K.: An algorithm for reducing the bandwidth and profile of a sparse matrix. SIAM J. Numer. Anal. 13(2), 236–250 (1976)CrossRefzbMATHGoogle Scholar
  15. 15.
    Everstine, G.C.: A comparison of three resequencing algorithms for the reduction of matrix profile and wavefront. Int. J. Numer. Methods. Eng. 14, 837–853 (1979)CrossRefzbMATHGoogle Scholar
  16. 16.
    Gibbs, N.E.: A hybrid profile reduction algorithm. ACM Trans. Math. Softw. 2(4), 378–387 (1976)CrossRefGoogle Scholar
  17. 17.
    Lewis, J.G.: Implementation of the gibbs-poole-stockmeyer and gibbs-king algorithms. ACM. Trans. Math. Softw. 8(2), 180–189 (1982)CrossRefzbMATHGoogle Scholar
  18. 18.
    Armstrong, B.A.: Near minimal matrix profiles and wavefronts for testing nodal resequencing algorithms. Int. J. Numer. Methods. Eng. 21, 1785–1790 (1985)CrossRefzbMATHGoogle Scholar
  19. 19.
    Sloan, S.W.: A FORTRAN program for profile and wavefront reduction. Int. J. Numer. Methods. Eng. 28(11), 2651–2679 (1989)CrossRefzbMATHGoogle Scholar
  20. 20.
    Sloan, S.W., Ng, W.S.: A direct comparison of three algorithms for reducing profile and wavefront. Comput. Struct. 33(2), 411–419 (1989)CrossRefGoogle Scholar
  21. 21.
    Duff, I., Reid, J.K., Scott, J.A.: The use of profile reduction algorithms with a frontal code. Int. J. Numer. Methods. Eng. 28(11), 2555–2568 (1989)CrossRefzbMATHGoogle Scholar
  22. 22.
    Kumfert, G., Pothen, A.: Two improved algorithms for envelope and wavefront reduction. BIT 37, 559–590 (1997)CrossRefzbMATHGoogle Scholar
  23. 23.
    Reid, J.K., Scott, J.A.: Ordering symmetric sparse matrices for small profile and wavefront. Int. J. Numer. Methods. Eng. 45, 1737–1755 (1999)CrossRefzbMATHGoogle Scholar
  24. 24.
    Barnard, S.T., Pothen, A., Simon, H.D.: A spectral algorithm for envelope reduction of sparse matrices. In: Proceedings of the Supercomputing ’93: Proceedings of the 1993 ACM/IEEE conference on Supercomputing, pp. 493–502. ACM, New York (1993).Google Scholar
  25. 25.
    Hu, Y.F., Scott, J.A.: A multilevel algorithm for wavefront reduction. SIAM J. Sci. Comput. 23(4), 1352–1375 (2001)CrossRefzbMATHGoogle Scholar
  26. 26.
    Hager, W.: Minimizing the profile of a symmetric matrix. SIAM J. Sci. Comput. 23(5), 1799–1816 (2002)CrossRefzbMATHGoogle Scholar
  27. 27.
    Reid, J.K., Scott, J.A.: Implementing hager’s exchange methods for matrix profile reduction. ACM Trans. Math. Softw. 28(4), 377–391 (2002)CrossRefzbMATHGoogle Scholar
  28. 28.
    Wang, Q., Shi, X.W.: An improved algorithm for matrix bandwidth and profile reduction in finite element analysis. Prog. Electromagn. Res. Lett. 9, 29–38 (2009)CrossRefGoogle Scholar
  29. 29.
    Marti, R., Laguna, M., Glover, F., Campos, V.: Reducing the bandwidth of a sparse matrix with tabu search. Eur. J. Oper. Res. 135(2), 450–459 (2001)CrossRefzbMATHGoogle Scholar
  30. 30.
    Lim, A., Rodrigues, B., Xiao, F.: Integrated genetic algorithm with hill climbing for bandwidth minimization problem. In: Proceedings of the 2003 international conference on Genetic and evolutionary computation: Part II, pp. 1594–1595. Springer-Verlag, Berlin (2003).Google Scholar
  31. 31.
    Piñana, E., Plana, I., Campos, V., Martí, R.: GRASP and path relinking for the matrix bandwidth minimization. Eur. J. Oper. Res. 153(1), 200–210 (2004)CrossRefzbMATHGoogle Scholar
  32. 32.
    Lim, A., Lin, J., Rodrigues, B., Xiao, F.: Ant colony optimization with hill climbing for the bandwidth minimization problem. Appl. Soft. Comput. 6(2), 180–188 (2006)CrossRefGoogle Scholar
  33. 33.
    Lim, A., Lin, J., Xiao, F.: Particle swarm optimization and hill climbing for the bandwidth minimization problem. Appl. Intell. 26(3), 175–182 (2007)CrossRefzbMATHGoogle Scholar
  34. 34.
    Rodriguez-Tello, E., Jin-Kao, H., Torres-Jimenez, J.: An improved simulated annealing algorithm for bandwidth minimization. Eur. J. Oper. Res. 185(3), 1319–1335 (2008)CrossRefzbMATHGoogle Scholar
  35. 35.
    Rodriguez-Tello, E., Hao, J.K., Torres-Jimenez, J.: An effective two-stage simulated annealing algorithm for the minimum linear arrangement problem. Comput. Oper. Res. 35(10), 3331–3346 (2008)CrossRefzbMATHGoogle Scholar
  36. 36.
    Ross, P.: Hyper-heuristics. In: Burke, E.K., Kendall, G. (eds.) Search Methodologies, pp. 529–556. Springer, New York (2005)CrossRefGoogle Scholar
  37. 37.
    de Abreu, N.M.M.: Old and new results on algebraic connectivity of graphs. Lin. Algebra. Appl. 423(1), 53–73 (2007)CrossRefzbMATHGoogle Scholar
  38. 38.
    Fiedler, M.: Algebraic connectivity of graphs. Czechoslovak. Math. J. 23, 298–305 (1973)Google Scholar
  39. 39.
    Lim, A., Rodrigues, B., Xiao, F.: Heuristics for matrix bandwidth reduction. Eur. J. Oper. Res. 174(1), 69–91 (2006)CrossRefzbMATHGoogle Scholar
  40. 40.
    Mladenovic, N., Urosevic, D., Prez-Brito, D., Garca-Gonzlez, C.G.: Variable neighbourhood search for bandwidth reduction. Eur. J. Oper. Res. 200(1), 14–27 (2010)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Engineering-Emerging TechnologiesUniversity of TabrizTabrizIran
  2. 2.School of Computer Science and Electronic EngineeringUniversity of EssexColchesterUK

Personalised recommendations