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The P Vs NP Problem: A Method for Solving NP-Complete Problems by use of Graph Embodiment on a Quantum Computation Paradigm using a Relational Database Query

Shadrack Mwanik, Andrew Mwaura Kahonge, Evans Kirimi Miriti

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

The relationship between the complexity class P and NP is one of the most fascinating and unresolved question in theoretical Computer Science. The classical computational paradigm, hedged on Turing thesis may be by itself the limiting factor. To investigate this relationship, a method for solving a query problem on a relational database on the classical and the quantum computational paradigm has been developed. The method solves queries or database problems through the use of graphs. The results indicate that the solution to the P vs NP cannot be resolved under the current computation paradigms and may requires a new computation paradigm to resolve it.

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The relationship between the complexity class P and NP is one of the most fascinating and unresolved question in theoretical Computer Science. The classical computational paradigm, hedged on Turing thesis may be by itself the limiting factor. To investigate this relationship, a method for solving a query problem on a relational database on the classical and the quantum computational paradigm has been developed. The method solves queries or database problems through the use of graphs. The results indicate that the solution to the P vs NP cannot be resolved under the current computation paradigms and may requires a new computation paradigm to resolve it.
The P vs. NP problem is a central open question in theoretical computer science, asking whether every problem whose solution can be verified in polynomial time can also be solved in polynomial time. This paper presents an interdisciplinary exploration of this problem by integrating mathematical analysis, quantum computing, and machine learning.We analyze the limitations of polynomial-time reductions among NP-complete problems under the Exponential Time Hypothesis (ETH), providing detailed proofs that offer new insights into the structural complexities within NP. Additionally, we develop a novel quantum-classical hybrid algorithm that combines Grover's algorithm with classical heuristics to solve specific instances of NP-complete problems more efficiently. This includes a comprehensive complexity analysis, scalability discussions, and detailed practical implementation aspects such as oracle constructions.Furthermore, we propose a machine learning framework based on Graph Neural Networks (GNNs) to capture patterns within NP-complete problems. This approach is theoretically justified, enhances heuristic methods, and demonstrates superior performance through extensive experimental comparisons with classical heuristics and exact solvers.By integrating these methods, this work offers fresh perspectives on the P vs. NP problem and contributes to the broader understanding of computational complexity.
The relevance of the research is determined by the fundamental nature of the P vs NP problem, which has been considered a central unsolved problem in theoretical computer science for more than half a century. In this article, for the first time, it is proved that the problem of solving the P vs NP problem is not a mathematical problem, but a physical problem, the solution of which depends on the physical implementation of the computing system. The aim of this paper is to prove that the solution of the P vs NP problem depends on the type of physical system used for calculations. The leading approach to research is to synthesize three independent paradigms: 1. Algebraic topology that provides invariants for complexity classification. 2. Quantum physics, which provides parallel information processing. 3. Apparat implementation on the GPU that demonstrates the physical limitations of classical computing. Author's results include: A formal proof that P ≠ NP holds in classical physics (Newtonian mechanics, deterministic computations) and a simultaneous proof that P = NP holds in quantum physics (superposition, entanglement). Experimental confirmation of the physical implementation through a series of computational experiments. Development of hybrid system that allows you to switch between modes depending on the available resources. The theoretical significance lies in rethinking the fundamental problem of complexity theory as a physical law. Practical significance is the creation of an adaptive computing system, the efficiency of which is determined by the available physical platform.
In this paper, we propose two new methods for solving Set Constraint Problems, as well as a potential polynomial solution for NP-Complete problems using quantum computation. While current methods of solving Set Constraint Problems focus on classical techniques, we offer both a quantum-inspired matrix method and a quantum matrix method that neutralizes common contradictions and inconsistencies that appear in these types of problems. We then use our new method to show how a potential polynomial solution for NP-Complete problems could be found using quantum computation. We state this as a potential solution, rather than an actual solution, as the outcome of any quantum computation may not be the same as the expected outcome. We start by formally defining a Set Constraint Problem. We then explain current, classical methods that are used to solve these problems and the drawbacks of such methods. After this, we explain a new quantum-inspired matrix method that allows us to solve these problems, with classical limitations. Then, we explain a new quantum matrix method that solves these problems using quantum information science. Finally, we describe how we can extend this method to potentially solve NP-Complete problems in polynomial time using quantum computation.
Can the computational complexity theory of computer science and mathematics say something new about unresolved problems in quantum physics? Particularly, can the P versus NP question in the computational complexity theory be a factor in the elucidation of the emergency of classicality in quantum mechanics? The paper compares two different ways of deriving classicality from the quantum formalism resulted from two differing hypotheses regarding the P versus NP question -- the approach of the quantum decoherence theory implying that P = NP and the computational complexity approach which assumes that P is not equal to NP.
The Subset Sum Problem is a member of the NP-complete class, so no known polynomial time algorithm exists for it. Although there are polynomial time approximations and heuristics, these are not always acceptable, yet exact-solution algorithms are unfeasible for large input. Quantum computation offers new insights for not only the Subset Sum Problem but also the entire NP-complete class; most notably, Grover's quantum algorithm for an unstructured database search can be tailored to identify solutions to problems within mathematics and computer science. This paper discusses the physical and conceptual feasibility of quantum computation and demonstrates the utility of quantum search by analyzing the time complexities of the classical dynamic programming algorithm and Grover's algorithm in solving the Subset Sum Problem, evincing the implications this has on the NP-complete class in general.