Skip to main content
Book cover

Optimization pp 227–243Cite as

Estimating the size of correcting codes using extremal graph problems

Part of the Springer Optimization and Its Applications book series (SOIA,volume 32)

Abstract

Some of the fundamental problems in coding theory can be formulated as extremal graph problems. Finding estimates of the size of correcting codes is important from both theoretical and practical perspectives. We solve the problem of finding the largest correcting codes using previously developed algorithms for optimization problems in graphs. We report new exact solutions and estimates.

Keywords

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Abello, S. Butenko, P. Pardalos and M. Resende, Finding independent sets in a graph using continuous multivariable polynomial formulations, J. Global Optim. 21(4) (2001), 111–137.

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Arora and S. Safra, Approximating clique is NP–complete, Proceedings of the 33rd IEEE Symposium on Foundations on Computer Science (1992) (IEEE Computer Society Press, Los Alamitos, California, 1992), 2–13.

    Google Scholar 

  3. I. M. Bomze, M. Budinich, P. M. Pardalos and M. Pelillo, The maximum clique problem, in D.-Z. Du and P. M. Pardalos, Eds, Handbook of Combinatorial Optimization (Kluwer Academic Publishers, Dordrecht, 1999), 1–74.

    Google Scholar 

  4. A. Brouwer, J. Shearer, N. Sloane and W. Smith, A new table of constant weight codes, IEEE Trans. Inform. Theory 36 (1990), 1334–1380.

    Article  MATH  MathSciNet  Google Scholar 

  5. S. Butenko, P. M. Pardalos, I. V. Sergienko, V. Shylo and P. Stetsyuk, Finding maximum independent sets in graphs arising from coding theory, Proceedings of the 17th ACM Symposium on Applied Computing (ACM Press, New York, 2002), 542–546.

    Google Scholar 

  6. S. D. Constantin and T. R. N. Rao, On the theory of binary asymmetric error correcting codes, Inform. Control 40 (1979), 20–36.

    Article  MATH  MathSciNet  Google Scholar 

  7. P. Delsarte and P. Piret, Bounds and constructions for binary asymmetric error correcting codes, IEEE Trans. Inform. Theory IT-27 (1981), 125–128.

    Article  MathSciNet  Google Scholar 

  8. I. I. Dikin, Iterative solution of linear and quadratic programming problems, Dokl. Akad. Nauk. SSSR 174 (1967), 747–748 (in Russian).

    MathSciNet  Google Scholar 

  9. I. I. Dikin and V. I. Zorkal’tsev, Iterative Solution of Mathematical Programming Problems (Algorithms for the Method of Interior Points) (Nauka, Novosibirsk, 1980).

    Google Scholar 

  10. J. Dongarra, C. Moler, J. Bunch and G. Stewart, Linpack users’ guide, http://www.netlib.org/linpack/index.html, available from the ICTP Library, 1979.

  11. T. Etzion, New lower bounds for asymmetric and undirectional codes, IEEE Trans. Inform. Theory 37 (1991), 1696–1704.

    Article  MATH  MathSciNet  Google Scholar 

  12. T. Etzion and P. R. J. Ostergard, Greedy and heuristic algorithms for codes and colorings, IEEE Trans. Inform. Theory 44 (1998), 382–388.

    Article  MATH  MathSciNet  Google Scholar 

  13. U. Feige and J. Kilian, Zero knowledge and the chromatic number, J. Comput. System Sci. 57 (1998), 187–199.

    Article  MATH  MathSciNet  Google Scholar 

  14. M. R. Garey and D. S. Johnson, The complexity of near–optimal coloring, JACM 23 (1976), 43–49.

    Article  MATH  MathSciNet  Google Scholar 

  15. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP–completeness (Freeman, San Francisco, 1979).

    MATH  Google Scholar 

  16. J. Håstad, Clique is hard to approximate within n 1–∈, Acta Math. 182 (1999), 105–142.

    Article  MATH  MathSciNet  Google Scholar 

  17. D. S. Johnson and M. A. Trick (Eds), Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, Vol. 26 of DIMACS Series, (American Mathematical Society, Providence, RI, 1996).

    MATH  Google Scholar 

  18. C. Lund and M. Yannakakis, On the hardness of approximating minimization problems, JACM 41 (1994), 960–981.

    Article  MATH  MathSciNet  Google Scholar 

  19. P. M. Pardalos, T. Mavridou and J. Xue, The graph coloring problem: a bibliographic survey, in D.-Z. Du and P. M. Pardalos, Eds, Handbook of Combinatorial Optimization, Vol. 2 (Kluwer Academic Publishers, Dordrecht, 1999), 331–395.

    Google Scholar 

  20. T. R. N. Rao and A. S. Chawla, Asymmetric error codes for some lsi semiconductor memories, Proceedings of the 7th Southeastern Symposium on System Theory (1975) (IEEE Computer Society Press, Los Alamitos, California, 1975), 170–171.

    Google Scholar 

  21. I. V. Sergienko, V. P. Shylo and P. I. Stetsyuk, Approximate algorithm for solving the maximum independent set problem, in Computer Mathematics, (V.M. Glushkov Institute of Cybernetics NAS of Ukraine, Kiev, 2001), 4–20 (in Russian).

    Google Scholar 

  22. V. Shylo, New lower bounds of the size of error–correcting codes for the Z–channel, Cybernet. Systems Anal. 38 (2002), 13–16.

    Article  MathSciNet  Google Scholar 

  23. V. Shylo and D. Boyarchuk, An algorithm for construction of covering by independent sets, in Computer Mathematics (V.M. Glushkov Institute of Cybernetics NAS of Ukraine, Kiev, 2001), 151–157.

    Google Scholar 

  24. N. Sloane, Challenge problems: Independent sets in graphs, http://www.research.att.com/njas/doc/graphs.html, 2001.

  25. N. Sloane, On single–deletion–correcting codes, in K. T. Arasu and A. Suress, Eds, Codes and Designs: Ray–Chaudhuri Festschrift (Walter de Gruyter, Berlin, 2002), 273–291.

    Google Scholar 

  26. C. L. M. van Pul and T. Etzion, New lower bounds for constant weight codes, IEEE Trans. Inform. Theory 35 (1989), 1324–1329.

    Article  MATH  Google Scholar 

  27. R. R. Varshamov, A class of codes for asymmetric channels and a problem from the additive theory of numbers, IEEE Trans. Inform. Theory IT–19 (1973), 92–95.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

We would like to thank two anonymous referees for their valuable comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sergiy Butenko .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag New York

About this chapter

Cite this chapter

Butenko, S., Pardalos, P., Sergienko, I., Shylo, V., Stetsyuk, P. (2009). Estimating the size of correcting codes using extremal graph problems. In: Pearce, C., Hunt, E. (eds) Optimization. Springer Optimization and Its Applications, vol 32. Springer, New York, NY. https://doi.org/10.1007/978-0-387-98096-6_12

Download citation

Publish with us

Policies and ethics