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$P$ 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 $\\textit{P-Sel}$. $\\textit{P-Sel}$ is the class of decision problems for which there is a polynomial time algorithm (called a selector) with the following property: Whenever it's given two instances, a "yes" and a "no" instance, the algorithm can always decide which is the "yes" instance. It is known that if $NP$ is contained in $\\textit{P-Sel}$, then $P = NP$.
\n\nIn this paper we consider the problem of computing the sum of the weighted densities of states of a Boolean formula in $3CNF$. Given a Boolean formula $\\phi$, the density of states $n(E)$ counts the number of truth assignments that leave exactly $E$ clauses unsatisfied in $\\phi$. The weighted density of states $m(E)$ is equal to $E \\times n(E)$. The sum of the weighted densities of states of a Boolean formula in $3CNF$ with $m$ clauses is equal to $\\sum_{E = 0}^{m} m(E)$. We prove that we can calculate the sum of the weighted densities of states in polynomial time. The lowest value of $E$ with a non-zero density (i.e. $min_{E}\\{E|n(E) > 0\\}$) is the solution of the corresponding $\\textit{MAX-SAT}$ problem.
\n\nThe minimum lowest value with a non-zero density from the two formulas $\\phi_{1}$ and $\\phi_{2}$ is equal to the minimum value between $E_{1}$ and $E_{2}$, where $E_{i}$ is the lowest value with a non-zero density of $\\phi_{i}$ for $i \\in \\{1, 2\\}$. Given two Boolean formulas $\\phi_{1}$ and $\\phi_{2}$ in $3CNF$ with $n$ variables and $m$ clauses, the combinatorial optimization problem $\\textit{SELECTOR-3SAT}$ consists in selecting the formula which has the minimum lowest value with a non-zero density, where every clause from $\\phi_{1}$ and $\\phi_{2}$ can be unsatisfied for some truth assignment. We assume that the formula with the minimum lowest value with a non-zero density has the minimum sum of the weighted densities of states. In this way, we solve $\\textit{SELECTOR-3SAT}$ with an exact polynomial time algorithm. We claim that this could be used for a possible selector of $3SAT$ and thus, $P = NP$.
", "publication_date": "2020-07-18", "publisher": "Zenodo", "resource_type": { "id": "publication-preprint", "title": { "de": "Preprint", "en": "Preprint" } }, "rights": [ { "description": { "en": "The Creative Commons Attribution license allows re-distribution and re-use of a licensed work on the condition that the creator is appropriately credited." }, "icon": "cc-by-icon", "id": "cc-by-4.0", "props": { "scheme": "spdx", "url": "https://creativecommons.org/licenses/by/4.0/legalcode" }, "title": { "en": "Creative Commons Attribution 4.0 International" } } ], "subjects": [ { "subject": "complexity classes" }, { "subject": "combinatorial optimization" }, { "subject": "reduction" }, { "subject": "polynomial time" }, { "subject": "logarithmic space" }, { "subject": "one-way" } ], "title": "P versus NP" }, "parent": { "access": { "owned_by": { "user": 45971 } }, "communities": {}, "id": "3355776", "pids": { "doi": { "client": "datacite", "identifier": "10.5281/zenodo.3355776", "provider": "datacite" } } }, "pids": { "doi": { "client": "datacite", "identifier": "10.5281/zenodo.3951064", "provider": "datacite" }, "oai": { "identifier": "oai:zenodo.org:3951064", "provider": "oai" } }, "revision_id": 32, "stats": { "all_versions": { "data_volume": 1621983041.0, "downloads": 3768, "unique_downloads": 3646, "unique_views": 3760, "views": 4343 }, "this_version": { "data_volume": 6453650.0, "downloads": 14, "unique_downloads": 14, "unique_views": 20, "views": 20 } }, "status": "published", "updated": "2021-02-10T00:10:50.989244+00:00", "versions": { "index": 104, "is_latest": false } }