Skip to main content

Advertisement

SpringerLink
Log in

A biased random-key genetic algorithm for the Steiner triple covering problem

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

We present a biased random-key genetic algorithm (BRKGA) for finding small covers of computationally difficult set covering problems that arise in computing the 1-width of incidence matrices of Steiner triple systems. Using a parallel implementation of the BRKGA, we compute improved covers for the two largest instances in a standard set of test problems used to evaluate solution procedures for this problem. The new covers for instances A 405 and A 729 have sizes 335 and 617, respectively. On all other smaller instances our algorithm consistently produces covers of optimal size.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aiex R.M., Resende M.G.C., Ribeiro C.C.: TTTPLOTS: a perl program to create time-to-target plots. Optim. Lett. 1, 201–212 (2007)

    Article  MathSciNet  Google Scholar 

  2. Avis D.: A note on some computationally difficult set covering problems. Math. Program. 18, 138–145 (1980)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  4. Feo T.A., Resende M.G.C.: A probabilistic heuristic for a computationally difficult set covering problem. Oper. Res. Lett. 8, 67–71 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fulkerson D.R., Nemhauser G.L., Trotter L.E. Jr.: Two computationally difficult set covering problems that arise in computing the 1-width of incidence matrices of Steiner triple systems. Math. Programm. Study 2, 72–81 (1974)

    Article  Google Scholar 

  6. Garey M.R., Johnson D.S.: Computers and intractability. A guide to the theory of NP-completeness. WH Freeman and Company, San Francisco (1979)

    MATH  Google Scholar 

  7. Gonçalves, J.F., Resende, M.G.C.: Biased random-key genetic algorithms for combinatorial optimization. J. Heuristics (2010a). doi:10.1007/s10732-010-9143-1

  8. Gonçalves, J.F., Resende, M.G.C.: A parallel multi-population genetic algorithm for a constrained two-dimensional orthogonal packing problem. J. Combin. Optim. (2010b). doi:10.1007/s10878-009-9282-1

  9. Hall M.: Combinatorial theory. Blaisdell Company, Minneapolis (1967)

    MATH  Google Scholar 

  10. Johnson D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9, 256–278 (1974)

    Article  MATH  Google Scholar 

  11. Karmarkar N.K., Resende M.G.C., Ramakrishnan K.G.: An interior point algorithm to solve computationally difficult set covering problems. Math. Programm. 52, 597–618 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mannino C., Sassano A.: Solving hard set covering problems. Oper. Res. Lett. 18, 1–5 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  13. Matsumoto M., Nishimura T.: Mersenne Twister: a 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Trans. Model. Comput. Simul. 8, 3–30 (1998)

    Article  MATH  Google Scholar 

  14. Odijk M.A., van Maaren H.: Improved solutions to the Steiner triple covering problem. Inf. Process. Lett. 65, 67–69 (1998)

    Article  Google Scholar 

  15. OpenMP. Last visited on 19 October (2010). http://openmp.org

  16. Östergård, P., Vaskelainen, V.: Russian doll search for the Steiner triple covering problem. Optimization Letters, pp. 1–8 (2010) doi:10.1007/s11590-010-0225-7

  17. Ostrowski, J., Linderoth, J., Rossi, F., Smriglio, S.: Solving large Steiner triple covering problems. Technical Report 1663, Computer Sciences Department, University of Wisconsin, Madison (2009)

  18. Ostrowski, J., Linderoth, J., Rossi, F., Smriglio, S.: Solving Steiner triple covering problems. Optima 83 (2010)

  19. Resende M.G.C., Toso R.F.: BRKGA framework: A C++ framework for implementing biased random-key genetic algorithms. Technical report, AT&T Labs Research, USA (2010)

    Google Scholar 

  20. Spears, W.M., DeJong, K.A.: On the virtues of parameterized uniform crossover. In: Proceedings of the Fourth International Conference on Genetic Algorithms, pp. 230–236 (1991)

  21. Stroustrup B.: The C++ Programming Language. Addison-Wesley, Reading (1997)

    Google Scholar 

  22. Vemuganti R.R.: Applications of set covering, set packing and set partitioning models: a survey. In: Du, D.-Z., Pardalos, P.M. (eds) Handbook of Combinatorial Optimization, vol. 1, pp. 573–746. Kluwer, Dordrecht (1998)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Algorithms and Optimization Research Department, AT&T Labs Research, 180 Park Avenue, Room C241, Florham Park, NJ, 07932, USA

    Mauricio G. C. Resende

  2. Department of Computer Science, Rutgers University, 110 Frelinghuysin Road, Piscataway, NJ, 08854-8019, USA

    Rodrigo F. Toso

  3. Faculdade de Economia do Porto, NIAAD, Rua Dr. Roberto Frias, 4200-464, Porto, Portugal

    José Fernando Gonçalves

  4. Centro de Informática (CIn), Federal University of Pernambuco, Av.Professor Luís Freire s/n, Cidade Universitária, Recife, PE, Brazil

    Ricardo M. A. Silva

Authors
  1. Mauricio G. C. Resende
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rodrigo F. Toso
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. José Fernando Gonçalves
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ricardo M. A. Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mauricio G. C. Resende.

Additional information

AT&T Labs Research Technical Report.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Resende, M.G.C., Toso, R.F., Gonçalves, J.F. et al. A biased random-key genetic algorithm for the Steiner triple covering problem. Optim Lett 6, 605–619 (2012). https://doi.org/10.1007/s11590-011-0285-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-011-0285-3

Keywords

Search

Navigation