P versus NP

EasyChair Preprint no. 3061, version 9

13 pagesDate: July 10, 2020

Abstract

$P$ versus $NP$ is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is $P$ equal to $NP$? It was essentially mentioned in 1955 from a letter written by John Nash to the United States National Security Agency. However, a precise statement of the $P$ versus $NP$ problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US 1,000,000 prize for the first correct solution. A major complexity class is $\textit{NP--complete}$, where $3SAT$ is a well-known $\textit{NP--complete}$ problem. Given a set $I$ of Boolean formulas in $3CNF$ of $n$ variables and $m$ clauses, the combinatorial optimization problem $\textit{SELECTOR--3SAT}$ consists in selecting the formula with more satisfying truth assignments, where every clause from the formulas in $I$ can be unsatisfied for some truth assignment. We prove there is an exact polynomial time algorithm for $\textit{SELECTOR--3SAT}$. As result, we obtain that the complexity class $P$ is equal to $NP$.

Keyphrases: completeness, complexity classes, logarithmic space, one-way, polynomial time, reduction