Abstract
Sparse matrices emerge in a number of problems in science and engineering. Typically the efficiency of solvers for such problems depends crucially on the distances between the first non-zero element in each row and the main diagonal of the problem’s matrix — a property assessed by a quantity called the size of the envelope of the matrix. This depends on the ordering of the variables (i.e., the order of the rows and columns in the matrix). So, some permutations of the variables may reduce the envelope size which in turn makes a problem easier to solve. However, finding the permutation that minimises the envelope size is an NP-complete problem. In this paper, we introduce a hyper-heuristic approach based on genetic programming for evolving envelope reduction algorithms. We evaluate the best of such evolved algorithms on a large set of standard benchmarks against two state-of-the-art algorithms from the literature and the best algorithm produced by a modified version of a previous hyper-heuristic introduced for a related problem. The new algorithm outperforms these methods by a wide margin, and it is also extremely efficient.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bader-El-Den, M.B., Poli, R.: A GP-based hyper-heuristic framework for evolving 3-SAT heuristics. In: Thierens, D., Beyer, H.G., Bongard, J., Branke, J., Clark, J.A., Cliff, D., Congdon, C.B., Deb, K., Doerr, B., Kovacs, T., Kumar, S., Miller, J.F., Moore, J., Neumann, F., Pelikan, M., Poli, R., Sastry, K., Stanley, K.O., Stutzle, T., Watson, R.A., Wegener, I. (eds.) GECCO 2007: Proceedings of the 9th Annual Conference on Genetic and Evolutionary Computation, vol. 2, pp. 1749–1749. ACM Press, London (2007)
Barnard, S.T., Pothen, A., Simon, H.D.: A spectral algorithm for envelope reduction of sparse matrices. In: Supercomputing 1993: Proceedings of the 1993 ACM/IEEE Conference on Supercomputing, pp. 493–502. ACM, New York (1993)
Barnard, S.T., Pothen, A., Simon, H.D.: A spectral algorithm for envelope reduction of sparse matrices, dedicated to william kahan and beresford parlett (1993), http://citeseer.ist.psu.edu/64928.html
Burke, E., Kendall, G., Newall, J., Hart, E., Ross, P., Schulenburg, S.: Hyper-Heuristics: An Emerging Direction in Modern Search Technology. In: Handbook of Metaheuristics. International Series in Operations Research & Management Science, ch. 16, pp. 457–474 (2003)
Cowling, P.I., Kendall, G., Soubeiga, E.: A Hyperheuristic Approach to Scheduling a Sales Summit. In: Burke, E., Erben, W. (eds.) PATAT 2000. LNCS, vol. 2079, pp. 176–190. Springer, Heidelberg (2001)
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)
Everstine, G.C.: A comparison of three resequencing algorithms for the reduction of matrix profile and wavefront. International Journal for Numerical Methods in Engineering 14, 837–853 (1979)
George, J.A.: Computer implementation of the finite element method. Ph.D. thesis, Stanford, CA, USA (1971)
Gibbs, N.E.: A hybrid profile reduction algorithm. ACM Transactions on Mathematical Software 2(4), 378–387 (1976)
Gibbs, N.E., Poole, W.G., Stockmeyer, P.K.: An algorithm for reducing the bandwidth and profile of a sparse matrix. SIAM Journal on Numerical Analysis 13(2), 236–250 (1976)
Keller, R.E., Poli, R.: Linear genetic programming of parsimonious metaheuristics. In: Srinivasan, D., Wang, L. (eds.) 2007 IEEE Congress on Evolutionary Computation, September 25-28, pp. 4508–4515. IEEE Computational Intelligence Society, IEEE Press, Singapore (2007)
Koohestani, B., Poli, R.: A hyper-heuristic approach to evolving algorithms for bandwidth reduction based on genetic programming. In: Bramer, M., Petridis, M., Nolle, L. (eds.) Research and Development in Intelligent Systems XXVIII, pp. 93–106. Springer, London (2011)
Koza, J.R.G.P.: On the Programming of Computers by Means of Natural Selection. MIT Press, Cambridge (1992)
Oltean, M.: Evolving evolutionary algorithms using linear genetic programming. Evolutionary Computation 13(3), 387–410 (Fall 2005)
Pissanetskey, S.: Sparse Matrix Technology. Academic Press, London (1984)
Poli, R., Langdon, W.B., McPhee, N.F.: A Field Guide to Genetic Programming (2008), published via http://lulu.com , with contributions by J. R. Koza
Poli, R.: Covariant tarpeian method for bloat control in genetic programming. In: Riolo, R., McConaghy, T., Vladislavleva, E. (eds.) Genetic Programming Theory and Practice VIII, Genetic and Evolutionary Computation, May 20-22, vol. 8, ch.5, pp. 71–90. Springer, Ann Arbor (2010)
Poli, R., Langdon, W.B., Holland, O.: Extending Particle Swarm Optimisation via Genetic Programming. In: Keijzer, M., Tettamanzi, A.G.B., Collet, P., van Hemert, J., Tomassini, M. (eds.) EuroGP 2005. LNCS, vol. 3447, pp. 291–300. Springer, Heidelberg (2005)
Poli, R., Woodward, J., Burke, E.K.: A histogram-matching approach to the evolution of bin-packing strategies. In: Srinivasan, D., Wang, L. (eds.) IEEE Congress on Evolutionary Computation, September 25-28, pp. 3500–3507. IEEE Computational Intelligence Society, IEEE Press, Singapore (2007)
Sloan, S.W.: A FORTRAN program for profile and wavefront reduction. International Journal for Numerical Methods in Engineering 28(11), 2651–2679 (1989)
Wang, Q., Shi, X.W.: An improved algorithm for matrix bandwidth and profile reduction in finite element analysis. Progress In Electromagnetics Research Letters 9, 29–38 (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Koohestani, B., Poli, R. (2012). A Genetic Programming Approach for Evolving Highly-Competitive General Algorithms for Envelope Reduction in Sparse Matrices. In: Coello, C.A.C., Cutello, V., Deb, K., Forrest, S., Nicosia, G., Pavone, M. (eds) Parallel Problem Solving from Nature - PPSN XII. PPSN 2012. Lecture Notes in Computer Science, vol 7492. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32964-7_29
Download citation
DOI: https://doi.org/10.1007/978-3-642-32964-7_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32963-0
Online ISBN: 978-3-642-32964-7
eBook Packages: Computer ScienceComputer Science (R0)