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 Zh. Vychisl. Mat. Mat. Fiz., 2011, Volume 51, Number 8, Pages 1531–1540 (Mi zvmmf9532)

Asymptotic estimates for the number of solutions of the dualization problem and its generalizations

E. V. Djukovaa, R. M. Sotnezovb

a Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333 Russia
b Moscow State University, Moscow, 119992 Russia

Abstract: Asymptotic estimates for the typical number of irreducible coverings and the typical length of an irreducible covering of a Boolean matrix are obtained in the case when the number $m$ of rows is no less than the number $n$ of columns. As a consequence, asymptotic estimates are obtained for the typical number of maximal conjunctions and the typical rank of a maximal conjunction of a monotone Boolean function of $n$ variables defined by a conjunctive normal form of $m$ clauses. Similar estimates are given for the number of irredundant coverings and the length of an irredundant covering of an integer matrix (for the number of maximal conjunctions and the rank of a maximal conjunction of a two-valued logical function defined by its zero set). Results obtained previously in this area are overviewed.

Key words: complexity of enumeration problems, dualization problem, maximal conjunction, irreducible covering of a Boolean matrix, irredundant covering of an integer matrix, complexity of search for irredundant coverings, metric properties of the set of coverings, metric properties of disjunctive normal forms, asymptotically optimal algorithm.

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English version:
Computational Mathematics and Mathematical Physics, 2011, 51:8, 1431–1440

Bibliographic databases:

UDC: 519.712
Revised: 18.01.2011

Citation: E. V. Djukova, R. M. Sotnezov, “Asymptotic estimates for the number of solutions of the dualization problem and its generalizations”, Zh. Vychisl. Mat. Mat. Fiz., 51:8 (2011), 1531–1540; Comput. Math. Math. Phys., 51:8 (2011), 1431–1440

Citation in format AMSBIB
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This publication is cited in the following articles:
1. E. V. Dyukova, R. M. Sotnezov, “On the complexity of the dualization problem”, Comput. Math. Math. Phys., 52:10 (2012), 1472–1481
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