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

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### 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.

## Keywords:

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

### ACKNOWLEDGMENTS

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

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