About a certain NP complete problem
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In this article, we introduce the concept of special decomposition of a set under the given pairs of subsets of that set, and the concept of special covering of a set under such a decomposition. We study the conditions for existence of special coverings of sets, under special decomposition of the set. Such conditions of formulated problem have important applications in the field of satisfiability of functions. Our goal is to study the relationship between sat CNF problem and the problem of existance of special covering of te set. We also study the relationship between classes of computational complexity by searching for special coverings of the sets. We prove, that the decidability of sat CNF problem, in polynomial time reduces to the problem of existence of a special covering of a set. We also prove, that the problem of existence of a special covering of a set, in polynomial time reduces to the decidability of the sat CNF problem. Therefore, the mentioned problems are polynomially equivalent. And then, the problem of existence of a special covering of a set is NP-complete problem.
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