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International Symposium on Stochastic Algorithms

SAGA 2005: Stochastic Algorithms: Foundations and Applications pp 1–13Cite as

Systems of Containers and Enumeration Problems

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3777))

Abstract

We discuss a technique (named “the container method”) for enumeration problems. It was applied for obtaining upper bounds and asymptotically sharp estimates for the number of independent sets, codes, antichains in posets, sum-free sets, monotone boolean functions and so on. The container method works even the appropriate recurrent equalities are absent and the traditional generating function method is not applicable. The idea of the method is to reduce a considered enumeration problem to evaluating the number of independent sets in the appropriate graph. We give some examples of such reduction and a survey of upper bounds for the number of independent sets in graphs. The method is usually successful if considered graphs are almost regular and expanders.

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References

  1. Alon, N.: Independent sets in regular graphs and Sum-Free Subsets of Finite Groups. Israel Journal of Math. 73(2), 247–256 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bollobas, B.: Random Graphs, 2nd edn., p. 495. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  3. Cameron, P., Erdős, P.: On the number of integers with various properties. In: Mollin, R.A. (ed.) Number Theory: Proc. First Conf. Can. Number Th. Ass., Banff, 1988, — de Gruyter, pp. 61–79 (1990)

    Google Scholar 

  4. Freiman, G.A.: Finite set addition. Izv. Vyssh. Uchebn. Zaved., Matematika 6(13), 202–213 (1959)

    MathSciNet  Google Scholar 

  5. Green, B.: The Cameron-Erdos conjecture. Bull. Lond. Math. Soc. 36(6), 769–778 (2004)

    Article  MATH  Google Scholar 

  6. Hansel, G.: Sur le nombre des fonctions booleen monotonesde n variables. C.R. Acad. Sci. Paris 262, 1088–1090 (1966)

    MathSciNet  Google Scholar 

  7. Katerinochkina, N.n.: Searcing of maximum upper zero of monotone Boolean functions. Doklady Acad. Nauk of URSS 224(3), 557–560 (1975) (in Russian)

    Google Scholar 

  8. Korobkov, V.K.: On monotone Boolean functions. Problemy Kibernetiki, M. Nauka 38, 5–108 (1981) (in Russian)

    Google Scholar 

  9. Korshunov, A.D.: On the number of monotone Boolean functions. Problemy Kibernetiki, M. Nauka 13, 5–28 (1965) (in Russian)

    Google Scholar 

  10. Korshunov, A.D., Sapozhenko, A.A.: On the number of binary codes with distance two. Problemy Kibernetiki, M. Nauka 40, 111–140 (1993) (in Russian)

    Google Scholar 

  11. Lev, V.F., Luczak, T., Schoen, T.: Sum-free sets in Abelian groups. Israel Journ. Math. 125(347), 347–367 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Sapozhenko, A.A.: On the number of connected sets with given size of boundary in graphs. Metody diskretnoi matematiki v reshenii combinatornykh zadach, - Novosibirsk 45, 35–58 (1987)

    Google Scholar 

  13. Sapozhenko, A.A.: On the number of antichains in posets. Discrete Mathematics and Applications 1(1), 35–59

    Google Scholar 

  14. Sapozhenko, A.A.: On the number of antichains in multylevelled posets. Discrete Mathematics and Applications 1(2), 149–171

    Google Scholar 

  15. Sapozhenko, A.A.: The Dedekind problem and boundary funcional method. Matematicheskie Voprosy Kibernetiki, M. Fizmatlit 9, 161–220 (2000)

    MathSciNet  Google Scholar 

  16. Sapozhenko, A.A.: On the Number of Independent Sets in Bipartite Graphs with Large Minimum Degree. DIMACS Technical Report 2000-25, 24–31 (August 2000) (in Russian)

    Google Scholar 

  17. Sapozhenko, A.A.: On the Number of Independent Sets in expanders. Diskretnaya matematika, Moscow 13(1), 56–62 (2001) (in Russian)

    Google Scholar 

  18. Sapozhenko, A.A.: On the number of sum-free sets in Abelian groups. Vestnik Moskovskogo Universiteta, ser. Math., Mech. 4, 14–18 (2002) (Russian)

    Google Scholar 

  19. Sapozhenko, A.A.: The Cameron-Erdos conjecture. Doklady of Russian Academy of Sciences 393(6), 749–752 (2003) (English Translation)

    Google Scholar 

  20. Sapozhenko, A.A.: On searcing upper zeroes of monotone functions on ranked posets. Journal of Mathematical Physics and Computational Mathematics 31(12), 1871–1884 (1991) (English translation)

    Google Scholar 

  21. Sapozhenko, A.A.: On the Number of Independent Sets in Graphs. Problems of theoretical cybernetics. In: Proceedings of XIII International Conference, Kazan, pp. 85–93, May 27-31 (2002)

    Google Scholar 

  22. Stepanov, V.E.: Phase transition in random graphs. Theory Probab. Applcs 15, 187–203

    Google Scholar 

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Author information

Authors and Affiliations

  1. Lomonosov University, Moscow

    Alexander Sapozhenko

Authors
  1. Alexander Sapozhenko
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Editor information

Editors and Affiliations

  1. Department of Discrete Mathematics, Leninskie Gory, Moscow State University, 119992, Moscow, Russia

    Oleg B. Lupanov  & Oktay M. Kasim-Zade  & 

  2. Faculty of Mechanics and Mathematics, Department of Discrete Mathematics, Leninskie Gory, Moscow State University, 119992, Moscow, Russia

    Alexander V. Chaskin

  3. Department of Computer Science, King’s College London, WC2R 2LS, London, UK

    Kathleen Steinhöfel

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© 2005 Springer-Verlag Berlin Heidelberg

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Sapozhenko, A. (2005). Systems of Containers and Enumeration Problems. In: Lupanov, O.B., Kasim-Zade, O.M., Chaskin, A.V., Steinhöfel, K. (eds) Stochastic Algorithms: Foundations and Applications. SAGA 2005. Lecture Notes in Computer Science, vol 3777. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11571155_1

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  • DOI: https://doi.org/10.1007/11571155_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-29498-6

  • Online ISBN: 978-3-540-32245-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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