Abstract
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.
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REFERENCES
E. V. Djukova, “On the complexity of implementation of some recognition procedures,” Zh. Vychisl. Mat. Mat. Fiz. 27, 114–127 (1987).
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].
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).
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).
E. V. Djukova, “On the Implementation Complexity of Discrete (Logical) Recognition Procedures,” Comput. Math. Math. Phys. 44, 532–541 (2004).
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).
D. S. Jonson, M. Yannakakis, and C. H. Papadimitriou, “On general all maximal independent sets,” Inf. Process. Lett. 27, 119–123 (1988).
M. L. Fredman and L. Khachiyan, “On the complexity of dualization of monotone disjunctive normal forms,” J. Algorithms 21, 618–628 (1996).
E. V. Djukova, “On an asymptotically optimal algorithm for constructing irredundant tests,” Dokl. Akad. Naul SSSR 233, 527–530 (1977).
E. V. Djukova, “Asymptotically optimal test algorithms in recognition problems,” Probl. Kibern., issue 39, (Nauka, Moscow, 1982), 165–199.
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].
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.
E. V. Djukova and P. A. Profjev, “Asymptotically optimal dualization algorithms,” Comput. Math. Math. Phys. 55, 891–916 (2015).
V. N. Noskov and V. A. Slepyan, “On the number of irredundant tests for a class of tables,” Kibernetica, No. 1, 60–65 (1972).
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).
ACKNOWLEDGMENTS
This work was supported by the Russian Foundation for Basic Research, project no. 16-01-00445-а.
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Translated by A. Klimontovich
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Djukova, E.V., Zhuravlev, Y.I. Monotone Dualization Problem and Its Generalizations: Asymptotic Estimates of the Number of Solutions. Comput. Math. and Math. Phys. 58, 2064–2077 (2018). https://doi.org/10.1134/S0965542518120102
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DOI: https://doi.org/10.1134/S0965542518120102