P versus NP

EasyChair Preprint 3061, version 10

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. Another major complexity class is $\textit{FNP--complete}$, where $\textit{MAX--3SAT}$ is a well-known $\textit{FNP--complete}$ problem. Given a set $I$ of Boolean formulas in $3CNF$ of $n$ variables and $m$ clauses, the combinatorial optimization problem $\textit{SELECTOR--MAX--3SAT}$ consists in selecting the formula with the truth assignment that satisfies the largest number of clauses between all the Boolean formulas in $I$, 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 $\textit{SELECTOR--MAX--3SAT}$. As result, we obtain that the complexity class $P$ is equal to $NP$.

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

@booklet{EasyChair:3061,
  author    = {Frank Vega},
  title     = {P versus NP},
  howpublished = {EasyChair Preprint 3061},
  year      = {EasyChair, 2020}}