On a Certain NP-Complete Problem

TL;DR

This paper introduces a new concept of special decompositions and coverings to analyze Boolean satisfiability, establishing their NP-completeness and exploring their potential for polynomial-time solutions.

Contribution

It defines special decompositions and coverings, proves their NP-completeness, and links these concepts to the SAT problem, aiming to find polynomial-time verification conditions.

Findings

Special covering existence is NP-complete.

Decidability of SAT reduces to special covering problem.

Problems are polynomially equivalent.

Abstract

We intend to create new concepts aimed at finding necessary and sufficient conditions for Boolean satisfiability so that these conditions can be verified in polynomial time. Based on these conditions it will be possible to create an algorithm that determines in polynomial time whether a given Boolean formula represented in conjunctive normal form is satisfiable. The work will consist of three articles. This is the first of a planned series of these articles. In this article we introduce the concept of special decomposition of a set and the concept of special covering for a set under such a decomposition. We formulate the decision problem of existance of a special covering for a set under a special decompostition of this set. In order to determine the complexity class in which this problem is located, we study the relationship between the sat CNF problem and the problem of existence of a special covering for a set under a special decomposition of this set. 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. The conditions for the existence of special coverings for a set under special decompositions of this set will be study in the next articles.