Note for the P versus NP Problem

Frank Vega

doi:10.20944/preprints201908.0037.v14

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preprints.org > computer science and mathematics > computer science > doi: 10.20944/preprints201908.0037.v14

Preprint Article Version 14 Preserved in Portico This version is not peer-reviewed

Note for the P versus NP Problem

Frank Vega

Version 1 : Received: 1 August 2019 / Approved: 5 August 2019 / Online: 5 August 2019 (03:31:23 CEST)
Version 2 : Received: 26 November 2019 / Approved: 29 November 2019 / Online: 29 November 2019 (07:28:14 CET)
Version 3 : Received: 4 April 2020 / Approved: 6 April 2020 / Online: 6 April 2020 (12:57:55 CEST)
Version 4 : Received: 15 April 2020 / Approved: 16 April 2020 / Online: 16 April 2020 (10:17:03 CEST)
Version 5 : Received: 18 September 2020 / Approved: 19 September 2020 / Online: 19 September 2020 (09:51:02 CEST)
Version 6 : Received: 10 February 2021 / Approved: 11 February 2021 / Online: 11 February 2021 (11:56:37 CET)
Version 7 : Received: 19 August 2021 / Approved: 27 August 2021 / Online: 27 August 2021 (14:09:47 CEST)
Version 8 : Received: 20 October 2021 / Approved: 26 October 2021 / Online: 26 October 2021 (11:07:59 CEST)
Version 9 : Received: 7 March 2024 / Approved: 8 March 2024 / Online: 8 March 2024 (11:06:11 CET)
Version 10 : Received: 14 March 2024 / Approved: 14 March 2024 / Online: 14 March 2024 (10:11:13 CET)
Version 11 : Received: 14 March 2024 / Approved: 15 March 2024 / Online: 18 March 2024 (08:29:48 CET)
Version 12 : Received: 29 March 2024 / Approved: 1 April 2024 / Online: 1 April 2024 (17:14:22 CEST)
Version 13 : Received: 6 April 2024 / Approved: 6 April 2024 / Online: 8 April 2024 (06:04:04 CEST)
Version 14 : Received: 11 April 2024 / Approved: 11 April 2024 / Online: 12 April 2024 (04:51:11 CEST)

How to cite: Vega, F. Note for the P versus NP Problem. Preprints 2019, 2019080037. https://doi.org/10.20944/preprints201908.0037.v14 Vega, F. Note for the P versus NP Problem. Preprints 2019, 2019080037. https://doi.org/10.20944/preprints201908.0037.v14

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MDPI and ACS Style

Vega, F. Note for the P versus NP Problem. Preprints 2019, 2019080037. https://doi.org/10.20944/preprints201908.0037.v14

APA Style

Vega, F. (2024). Note for the P versus NP Problem. Preprints. https://doi.org/10.20944/preprints201908.0037.v14

Chicago/Turabian Style

Vega, F. 2024 "Note for the P versus NP Problem" Preprints. https://doi.org/10.20944/preprints201908.0037.v14

Abstract

P versus NP is considered as one of the most fundamental 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. Another major complexity class is NP-complete. It is well-known that P is equal to NP under the assumption of the existence of a polynomial time algorithm for some NP-complete. We show that the Monotone Weighted Xor 2-satisfiability problem (MWX2SAT) is NP-complete and P at the same time. Certainly, we make a polynomial time reduction from every directed graph and positive integer k in the K-CLOSURE problem to an instance of MWX2SAT. In this way, we show that MWX2SAT is also an NP-complete problem. Moreover, we create and implement a polynomial time algorithm which decides the instances of MWX2SAT. Consequently, we prove that P = NP.

Keywords

Complexity classes; boolean formula; graph; completeness; polynomial time

Subject

Computer Science and Mathematics, Computer Science

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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