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### Solution of the Linear Ordering Problem (NP=P) [article]

Givi Bolotashvili
2003 arXiv   pre-print
A polynomial algorithm is obtained for the NP-complete linear ordering problem.  ...  Solution of the Linear Ordering Problem (NP=P) G.  ...  We denote the corresponding polytope by B n . The polytope B n has integer vertices corresponding to feasible solutions of the linear ordering problem as well as non-integer vertices.  ...

### P = NP [article]

Sergey V. Yakhontov
2017 arXiv   pre-print
The present work proves that P=NP.  ...  In more detail, machine M_AcceptingPath uses a reduction of the initial string problem to another string problem TCPE (defined in the paper) that is NP-complete and decidable in polynomial time.  ...  The shortened formulation of the problem is P =? NP. The papers      contain detailed surveys on the P vs. NP Problem. The present paper proves that P = NP.  ...

### P=NP [article]

Selmer Bringsjord, Joshua Taylor
2004 arXiv   pre-print
We claim to resolve the P=?NP problem via a formal argument for P=NP.  ...  Building the structure and the solution (and the container for the solution) requires steps linear in the size of n, and dipping and withdrawing make two steps, so despite the fact that STP is NP-complete  ...  of the 1000 or so currently catalogued NP-complete problems.  ...

### P versus NP on NP-completeness

Frank Vega
2017 Figshare
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?  ...  To attack the P versus NP question the concept of NP-completeness has been very useful. We prove some interesting results regarding this class which affect directly to the P versus NP problem.  ...  On the other hand, the OTHER-L is a problem of deciding given a solution y of an instance x in a language L whether x has another solution different of y.  ...

### P vs. NP [article]

Daniel Uribe
2016 arXiv   pre-print
Finally, the lower bound of such decision trees is demonstrated to not be in P.  ...  This method is then extended to optimal algorithms which may find all cliques of size q in network N, or simply the first clique of size q in network N.  ...  Informal Definition of NP-complete Informally, we may refer to NP-complete as the set of problems which are in NP but are possibly not in P.  ...

### P Versus Np

Frank Vega
2018 Zenodo
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?  ...  Moreover, we demonstrate this problem is NP-complete. To attack the P versus NP question the concept of NP-completeness has been very useful.  ...  Certainly, P versus N P is one of the greatest open problems in science and a correct solution for this incognita will have a great impact not only for computer science, but for many other fields as well  ...

### Philosophical Solution to P=?NP: P is Equal to NP [article]

Steven Meyer
2016 arXiv   pre-print
The P=?NP problem is philosophically solved by showing P is equal to NP in the random access with unit multiply (MRAM) model.  ...  Since the computation power of MRAMs is the same as NDTMs, P is equal to NP. The paper shows that the justification for the NDTM P?  ...  The paper also defends Deolaliker'ss cientific solution to the P=?NP problem where he shows P not equal to NP using properties of the statistical physics of Boltzmann gases.  ...

### P versus NP

Frank Vega
2020 Zenodo
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?  ...  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.  ...  We show some results that prove this outstanding problem with the unexpected solution of P = N P . 2 Theory and Methods Preliminaries In 1936, Turing developed his theoretical computational model  ...

### P versus NP

Frank Vega
2020 Zenodo
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?  ...  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.  ...  We show some results that prove this outstanding problem with the unexpected solution of P = N P . 2 Theory and Methods Preliminaries In 1936, Turing developed his theoretical computational model  ...

### P versus NP

Frank Vega
2020 Zenodo
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?  ...  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.  ...  We show some results that prove this outstanding problem with the unexpected solution of P = N P . 2 Theory and Methods Preliminaries In 1936, Turing developed his theoretical computational model  ...

### P versus NP

Frank Vega
2020 Zenodo
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?  ...  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.  ...  We show some results that prove this outstanding problem with the unexpected solution of P = N P . 2 Theory and Methods Preliminaries In 1936, Turing developed his theoretical computational model  ...

### A note on xQx as a modelling and solution framework for the Linear Ordering Problem

Mark Lewis, Bahram Alidaee, Fred Glover, Gary Kochenberger
2009 International Journal of Operational Research
In this paper, we demonstrate that the Linear Ordering Problem (LOP) can be easily recast so that it can be treated as a UQP problem, and that large instances of the LOP can be effectively handled within  ...  This paper expands the list of 0-1 problems that can be effectively modelled and solved as Unconstrained Quadratic Binary Programs (UQPs).  ...  Acknowledgement We thank the anonymous reviewers for their careful attention to detail and helpful comments.  ...

### P versus NP

Frank Vega
2019 Zenodo
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?  ...  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.  ...  We show some results that prove this outstanding problem with the unexpected solution of P = N P . 2 Theory and Methods Preliminaries In 1936, Turing developed his theoretical computational model  ...

### P versus NP

Frank Vega
2019 Zenodo
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?  ...  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.  ...  The following consequences are assuming that we have a practical solution for the NP-complete problems where such existence was proven with our nonconstructive result: 1.  ...

### P versus NP

Frank Vega
2019 Zenodo
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?  ...  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.  ...  The following consequences are assuming that we have a practical solution for the NP-complete problems where such existence was proven with our nonconstructive result: 1.  ...
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