Covering codes and combinatorial optimization

  • Patrick Solé
Submitted Contributions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 539)


It was proved by Ntafos and Hakimi in 1981 (and rediscovered recently by T. Zaslavsky and the author) that cycle codes of graphs could be completely decoded in polynomial time, by reduction to the Chinese Postman problem, and use of the Edmonds and Johnson algorithm. Upper and lower bounds on the covering radius of these codes were derived by the same authors. Shortly thereafter, A. Frank proved, using matching theory that the covering radius of these codes can also be computed in polynomial time. We report on these results as well as other results of the same type concerning cocycle codes of graphs. They are dual of the former and generalize the Gale-Berlekamp switching game.

We generalize the bounds on the covering radius of the cycle code of graphs to the cycle code of matroids having the sum of circuits property. This class of matroids, introduced by Seymour, contains the graphic matroids and certain cographic matroids as special cases. The associated codes can be completely decoded in polynomial time. The complexity of computing their covering radius is still unknown.


Polynomial Time Covering Radius Matching Theory Binary Linear Code Cycle Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    F. Barahona, A.R. Majhoub, “On the cut polytope”, Math. Programming 36 (1986), pp. 157–173.Google Scholar
  2. [2]
    C. Berge, Graphes, Masson (1984).Google Scholar
  3. [3]
    C. Berge, Hypergraphes, Masson (1987).Google Scholar
  4. [4]
    A.R. Calderbank, “Covering Radius and the Chromatic Number of Kneser Graphs”, J. of Comb. Th. A, 54, 1 (1990) 129–131.Google Scholar
  5. [5]
    J. Edmonds, E.L. Johnson, “Matching, Euler Tours and the Chinese Postman”, Math. Programming, 5 (1973), 88–124.Google Scholar
  6. [6]
    A. Frank, “Conservative weightings and ear-decomposition of graphs” submitted to Combinatorica.Google Scholar
  7. [7]
    R.L. Graham, N.J.A. Sloane, “On the covering radius of codes”, IEEE IT-31, pp. 385–401 (1985).Google Scholar
  8. [8]
    P.C. Fishburn, N.J.A. Sloane, “The solution to Gale-Berlekamp 's switching game.” Discr. Math. 74 (1989) 263–290.Google Scholar
  9. [9]
    M. Grötschel, K. Truemper, “Decomposition and Optimization over cycles in Binary Matroids” J. of Comb. Th. B 46, 306–337 (1989).Google Scholar
  10. [10]
    F. Harary, Graph Theory, Addison-Wesley (1969).Google Scholar
  11. [11]
    H. Janwa, “Some new upper bounds on the covering radius of binary linear codes”, IEEE Trans. on Inf. Th., IT-35, 110–122 (1989).Google Scholar
  12. [12]
    L. Lovasz, “On the ratio of optimal integral and fractionnal cover”, Discr, Math., 13,pp. 383–390 (1975).Google Scholar
  13. [13]
    A. McLoughlin, “The complexity of computing the covering radius of a code”, IEEE Trans. on Inform. Th., IT-30, 6, Nov. 84.Google Scholar
  14. [14]
    J. Bruck, M. Naor, “The hardness of decoding linear codes with preprocessing”, IBM Almaden Res. Report RJ 6504 (1988).Google Scholar
  15. [15]
    F.J. MacWilliams, N.J.A Sloane,The Theory of Error Correcting Codes,North-Holland (1981).Google Scholar
  16. [16]
    S.C. Ntafos, S.L. Hakimi, “On the complexity of some coding problems”, IEEE Trans. on Inform. Th., IT-27, 6, (1981) 794–796.Google Scholar
  17. [17]
    J. Pach, J. Spencer, “Explicit codes with a low covering radius”, IEEE Trans. on Inform. Th., IT-34, 5, Sept. 88.Google Scholar
  18. [18]
    A. Sebö, “The cographic multiflow problem: an epilogue”, IMAG Res. Report 808-M, February (1990).Google Scholar
  19. [19]
    P.D. Seymour, “The matroids with the max-flow min-cut property”, J. of Comb. Th. B, 189–222 (1977).Google Scholar
  20. [20]
    P.D. Seymour, “Decomposition of Regular Matroids”, J. of Comb. Th. B, 28 305–359(198).Google Scholar
  21. [21]
    P.D. Seymour, “Matroids and Multicommodity Flows”, European J. of Comb.2 (1981).Google Scholar
  22. [22]
    P. Solé, T. Zaslavsky, “Covering radius and Maximality of the cycle code of a graph” submitted to JCT B.Google Scholar
  23. [23]
    P. Solé, T. Zaslavsky, “A coding approach to signed graphs”. submitted to SIAM J. of Discr. Math.Google Scholar
  24. [24]
    D. Welsh, Matroid Theory, Academic Press (1976).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Patrick Solé
    • 1
  1. 1.CNRS, I3S, 250, rue Albert EinsteinValbonneFrance

Personalised recommendations