Estimating the size of correcting codes using extremal graph problems

Springer Optimization and Its Applications

Volumn 32, Issue , 2009, Pages 227-243

  Butenko, Sergiy ^a   Pardalos, Panos ^a   Sergienko, Ivan ^b   Shylo, Vladimir ^b   Stetsyuk, Petro ^b  

^a UNIVERSITY OF FLORIDA   (United States)

^b V M GLUSHKOV INSTITUTE OF CYBERNETICS   (Ukraine)

Author keywords

Coding theory; Combinatorial optimization; Error correcting codes; Graph coloring; Maximum independent set

Indexed keywords

EID: 84976481209     PISSN: 19316828     EISSN: 19316836     Source Type: Book Series    
DOI: 10.1007/978-0-387-98096-6_12     Document Type: Chapter

References (27)

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.
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.
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.
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.
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.
6. S. D. Constantin and T. R. N. Rao, On the theory of binary asymmetric error correcting codes, Inform. Control 40 (1979), 20-36.
7. P. Delsarte and P. Piret, Bounds and constructions for binary asymmetric error correcting codes, IEEE Trans. Inform. Theory IT-27 (1981), 125-128.
8. I. I. Dikin, Iterative solution of linear and quadratic programming problems, Dokl. Akad. Nauk. SSSR 174 (1967), 747-748 (in Russian).
9. I. I. Dikin and V. I. Zorkaltsev, Iterative Solution of Mathematical Programming Problems (Algorithms for the Method of Interior Points) (Nauka, Novosibirsk, 1980).
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.
12. T. Etzion and P. R. J. Ostergard, Greedy and heuristic algorithms for codes and colorings, IEEE Trans. Inform. Theory 44 (1998), 382-388.
13. U. Feige and J. Kilian, Zero knowledge and the chromatic number, J. Comput. System Sci. 57 (1998), 187-199.
14. M. R. Garey and D. S. Johnson, The complexity of near-optimal coloring, JACM 23 (1976), 43-49.
15. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness (Freeman, San Francisco, 1979).
16. J. Håstad, Clique is hard to approximate within n1-+, Acta Math. 182 (1999), 105-142.
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).
18. C. Lund and M. Yannakakis, On the hardness of approximating minimization problems, JACM 41 (1994), 960-981.
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.
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.
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).
22. V. Shylo, New lower bounds of the size of error-correcting codes for the Z-channel, Cybernet. Systems Anal. 38 (2002), 13-16.
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.
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.
26. C. L. M. van Pul and T. Etzion, New lower bounds for constant weight codes, IEEE Trans. Inform. Theory 35 (1989), 1324-1329.
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.