Decoding P vs. NP: A Multidimensional Exploration

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.