On a Certain NP-Complete Problem

In this article we introduce the concept of special decomposition of a set and the concept of special covering of a set under such a decomposition. Our goal is to study the conditions for the existence of special coverings for sets under special decompositions of these sets, as well as to study the degree of complexity of finding them. These conditions for the formulated problem will have important applications in the field of satisfiability of Boolean functions. In order to determine the complexity class in which this problem is located, we study the relationship between sat CNF problem and the problem of existence of special covering for the set. We also will study the relationship between classes of computational complexity by searching for special coverings of the sets. The article proves that the decidability of the sat CNF problem is polynomially reduced to the problem of the existence of a special covering for a set. It is also proved that the problem of existence of a special covering for a set is polynomially reduced 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 for a set is an NP-complete problem.