Advertisement

Optimization pp 227-243 | Cite as

Estimating the size of correcting codes using extremal graph problems

  • Sergiy ButenkoEmail author
  • Panos Pardalos
  • Ivan Sergienko
  • Vladimir Shylo
  • Petro Stetsyuk
Chapter
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

Maximum independent set graph coloring error-correcting codes coding theory combinatorial optimization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

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

References

  1. 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.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 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. 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. 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.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 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. 6.
    S. D. Constantin and T. R. N. Rao, On the theory of binary asymmetric error correcting codes, Inform. Control 40 (1979), 20–36.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    P. Delsarte and P. Piret, Bounds and constructions for binary asymmetric error correcting codes, IEEE Trans. Inform. Theory IT-27 (1981), 125–128.CrossRefMathSciNetGoogle Scholar
  8. 8.
    I. I. Dikin, Iterative solution of linear and quadratic programming problems, Dokl. Akad. Nauk. SSSR 174 (1967), 747–748 (in Russian).MathSciNetGoogle Scholar
  9. 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. 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. 11.
    T. Etzion, New lower bounds for asymmetric and undirectional codes, IEEE Trans. Inform. Theory 37 (1991), 1696–1704.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    T. Etzion and P. R. J. Ostergard, Greedy and heuristic algorithms for codes and colorings, IEEE Trans. Inform. Theory 44 (1998), 382–388.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    U. Feige and J. Kilian, Zero knowledge and the chromatic number, J. Comput. System Sci. 57 (1998), 187–199.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    M. R. Garey and D. S. Johnson, The complexity of near–optimal coloring, JACM 23 (1976), 43–49.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP–completeness (Freeman, San Francisco, 1979).zbMATHGoogle Scholar
  16. 16.
    J. Håstad, Clique is hard to approximate within n 1–∈, Acta Math. 182 (1999), 105–142.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 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).zbMATHGoogle Scholar
  18. 18.
    C. Lund and M. Yannakakis, On the hardness of approximating minimization problems, JACM 41 (1994), 960–981.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 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. 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. 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. 22.
    V. Shylo, New lower bounds of the size of error–correcting codes for the Z–channel, Cybernet. Systems Anal. 38 (2002), 13–16.CrossRefMathSciNetGoogle Scholar
  23. 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. 24.
    N. Sloane, Challenge problems: Independent sets in graphs, http://www.research.att.com/njas/doc/graphs.html, 2001.
  25. 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. 26.
    C. L. M. van Pul and T. Etzion, New lower bounds for constant weight codes, IEEE Trans. Inform. Theory 35 (1989), 1324–1329.zbMATHCrossRefGoogle Scholar
  27. 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.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  • Sergiy Butenko
    • 1
    Email author
  • Panos Pardalos
    • 1
  • Ivan Sergienko
    • 2
  • Vladimir Shylo
    • 2
  • Petro Stetsyuk
    • 2
  1. 1.University of FloridaGainesvilleUSA
  2. 2.Institute of CyberneticsNAS of UkraineKievUkraine

Personalised recommendations