Graph-Based Deterministic Polynomial Algorithm for NP Problems
Changryeol Lee
TL;DR
The work proposes a novel Computation Model and Feasible Graph framework to deterministically simulate NP verifiers in polynomial time, aiming to resolve P vs NP. It builds a dynamic computation graph representing all certificate verifications, then prunes infeasible paths to obtain a feasible graph that preserves all valid computation walks to final edges. A polynomial-time simulation algorithm is developed to decide NP problems by extending footmarks and exploring only necessary feasible paths, claiming P = NP. The approach carries profound theoretical implications for complexity theory and practical fields such as cryptography and optimization, while acknowledging challenges around practical implementability and resource requirements.
Abstract
The P = NP problem asks whether every problem whose solution can be verified in polynomial time (NP) can also be solved in polynomial time (P). In this paper, we present a proof that P = NP, demonstrating that every NP problem can be solved deterministically in polynomial time. We introduce a new Computation Model that enables the simulation of a Turing machine, and show that NP problems can be simulated efficiently within this framework. By introducing the concept of a Feasible Graph, we ensure that the simulation can be performed in polynomial time, providing a direct path to resolving the P = NP question. Our result has significant implications for fields such as cryptography, optimization, and artificial intelligence, where NP-complete problems play a central role.