Stephen J. Hartley
ABSTRACT
Steiner systems, particularly triple systems, are usually generated by mathematicians using techniques from the theory of groups and quasi-groups. When pencil-and-paper enumeration becomes infeasible, mathematicians have used computers to carry out exhaustive searches. This paper presents some results of using genetic algorithms, which do not use exhaustive search, to generate Steiner systems. A specialized mutation operator was effective in generating Steiner triple systems. Future research will focus on improving the genetic algorithm to generate higher order Steiner systems whose existence is not currently known.
- 1.C. J. Colbourn and P. C. van Oorschot, "Applications of Combinatorial Designs in Computer Science," A CM Computing Surveys, Vol. 21, No. 2, June 1989, pp. 223- 250. Google Scholar
Digital Library
- 2.K. D. Crawford, "Solving the n-Queens Problem Using Genetic Algorithms," Proceedings of the 1992 ACM/SIGAPP Syrup. on Applied Computing, Kansas City, MO, March 1-3, 1992, pp. 1039-1047. Google ScholarDigital Library
- 3.L. Davis, Handbook o.f Genetic Algorithms, Van Noetrand Reinhold, 1991.Google Scholar
- 4.K. A. DeJong and W. M. Spears, "Using Genetic Algorithms to Solve NP-Complete Problems," Proceedings of the 3rd International Conference on Genetic Algorithms, San Mateo, CA, June 4-7, 1989, pp. 124-132. Google ScholarDigital Library
- 5.J. Doyen and A. Rosa, "A Updated Bibliography and Survey of Steiner Systems," Ann. Discrete Math., Vol. 7, 1980, pp. 317-349.Google ScholarCross Ref
- 6.D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison Wesley, 1989. Google ScholarDigital Library
- 7.D. E. Goldberg, SGA-C (v1.1)- A Simple Genetic Algorithm, 1986, C version by Robert E. Smith, Univ. of Alabama, v l.1 modifications by Jeff Eaxickson, Boeing Company.Google Scholar
- 8.M. Hal!, Combinatorial Theory, Blaisdell Publishing, 1967.Google Scholar
- 9.J. H. Holland, Adaptation in Natural and Artificial Systems, Univ. of Michigan Press, Ann Arbor, MI, 1975. Google ScholarDigital Library
- 10.A. H. Konstam, S. J. Hartley, and W. L. Cart, "Optimization in a Distributed Processing Environment us. ing Genetic Algorithms with Multivariate Crossover," Proceedings of the 20th Annum ACM Computer Science Conference, Kansas City, MO, March 3-5, 1992, pp. 109-116. Google ScholarDigital Library
- 11.N. S. Mendelsohn and S. H. Y. Hung, "On the Steiner Systems S(3,4,14) and S(4,5,15)," Utilitas Mathematica, Vo}. 1, 1972, pp. 5-95.Google Scholar
- 12.J. Sch6nheim, "On Coverings," Pacific Journal of Mathematics, Vol. 14, 1964, pp. 1405-1411.Google ScholarCross Ref
Index Terms
- Using genetic algorithms to generate Steiner triple systems
Recommendations
Embedding Steiner triple systems in hexagon triple systems
A hexagon triple is the graph consisting of the three triangles (triples) {a,b,c},{c,d,e}, and {e,f,a}, where a,b,c,d,e, and f are distinct. The triple {a,c,e} is called an inside triple. A hexagon triple system of order n is a pair (X,H) where H is a ...
Caps and Colouring Steiner Triple Systems
Hill [6] showed that the largest cap in PG(5,3) has cardinality 56. Using this cap it is easy to construct a cap of cardinality 45 in AG(5,3). Here we show that the size of a cap in AG(5,3) is bounded above by 48. We also give an example of three disjoint ...
Automorphisms of steiner triple systems
This paper treats the following problem in combinatorial analysis: Find an incomplete balanced block design D with parameters b, v, r, k, and λ = 1, possessing an automorphism group G which is doubly transitive on the elements of D and such that the ...
Comments