Computational Mathematics and Mathematical Physics

, Volume 58, Issue 12, pp 2064–2077 | Cite as

Monotone Dualization Problem and Its Generalizations: Asymptotic Estimates of the Number of Solutions

  • E. V. DjukovaEmail author
  • Yu. I. ZhuravlevEmail author


Issues related to the construction of efficient algorithms for intractable discrete problems are studied. Enumeration problems are considered. Their intractability has two aspects—exponential growth of the number of their solutions with increasing problem size and the complexity of finding (enumerating) these solutions. The basic enumeration problem is the dualization of a monotone conjunctive normal form or the equivalent problem of finding irreducible coverings of Boolean matrices. For the latter problem and its generalization for the case of integer matrices, asymptotics for the typical number of solutions are obtained. These estimates are required, in particular, to prove the existence of asymptotically optimal algorithms for monotone dualization and its generalizations.


intractable discrete problem dualization of monotone conjunctive normal form irreducible covering of Boolean matrix irredundant covering of integer matrix asymptotically optimal algorithm 



This work was supported by the Russian Foundation for Basic Research, project no. 16-01-00445-а.


  1. 1.
    E. V. Djukova, “On the complexity of implementation of some recognition procedures,” Zh. Vychisl. Mat. Mat. Fiz. 27, 114–127 (1987).MathSciNetGoogle Scholar
  2. 2.
    E. V. Djukova, “Kora-type recognition algorithms: Implementation complexity and metric properties,” in Recognition, Classification, and Prediction (Mathematical Methods and Their Application) (Nauka, Moscow, 1989), No. 2, pp. 99–125 [in Russian].Google Scholar
  3. 3.
    E. V. Djukova, “Asymptotic estimates of certain characteristics of the set of representative collections and the stability problem,” Zh. Vychisl. Mat. Mat. Fiz. 35, 123–134 (1995).Google Scholar
  4. 4.
    E. V. Dyukova and Yu. I. Zhuravlev, “Discrete analysis of feature descriptions in recognition problems of high dimensionality,” Comput. Math. Math. Phys. 40, 1214–1227 (2000).MathSciNetzbMATHGoogle Scholar
  5. 5.
    E. V. Djukova, “On the Implementation Complexity of Discrete (Logical) Recognition Procedures,” Comput. Math. Math. Phys. 44, 532–541 (2004).MathSciNetGoogle Scholar
  6. 6.
    E. V. Djukova and V. Yu. Nefedov, “The complexity of transformation of normal forms for characteristic functions of classes,” Pattern Recognit. Image Anal. 19, 435–440 (2009).CrossRefGoogle Scholar
  7. 7.
    D. S. Jonson, M. Yannakakis, and C. H. Papadimitriou, “On general all maximal independent sets,” Inf. Process. Lett. 27, 119–123 (1988).CrossRefGoogle Scholar
  8. 8.
    M. L. Fredman and L. Khachiyan, “On the complexity of dualization of monotone disjunctive normal forms,” J. Algorithms 21, 618–628 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    E. V. Djukova, “On an asymptotically optimal algorithm for constructing irredundant tests,” Dokl. Akad. Naul SSSR 233, 527–530 (1977).MathSciNetGoogle Scholar
  10. 10.
    E. V. Djukova, “Asymptotically optimal test algorithms in recognition problems,” Probl. Kibern., issue 39, (Nauka, Moscow, 1982), 165–199.Google Scholar
  11. 11.
    E. V. Djukova and A. S. Inyakin, “Asymptotically optimal construction of irredundant coverings of integer matrices,” in Mathematical Problems of Cybernetics (Nauka, Moscow, 2008), No. 17, pp. 235–246 [in Russian].Google Scholar
  12. 12.
    E. V. Djukova and P. A. Profjev, “On the asymptotically optimal enumeration of irreducible coverings of Boolean matrices,” Prikl. Diskr. Mat., No. 1(23), 96–105.Google Scholar
  13. 13.
    E. V. Djukova and P. A. Profjev, “Asymptotically optimal dualization algorithms,” Comput. Math. Math. Phys. 55, 891–916 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    V. N. Noskov and V. A. Slepyan, “On the number of irredundant tests for a class of tables,” Kibernetica, No. 1, 60–65 (1972).MathSciNetGoogle Scholar
  15. 15.
    A. E. Andreev, “On the asymptotical behavior of the number of irredundant tests and the length of the minimal test for almost all tables,” Probl. Kibern., No. 41, 117–142 (1984).Google Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Dorodnicyn Computing Center, Federal Research Center “Computer Science and Control,” Russian Academy of SciencesMoscowRussia

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