Graph-Based Deterministic Polynomial Algorithm for NP Problems

The P versus NP problem asks whether every problem in NP, whose membership can be verified in polynomial time given a suitable certificate, can be decided by a deterministic Turing machine in polynomial time. In this paper, we present a proof that P = NP by constructing a deterministic polynomial-time algorithm for NP problems based on a graph-based computation framework.
We introduce a structured computation model in which the transitions of a deterministic Turing machine are incrementally realized in the corresponding computation graph via edge extensions. Each extension step enforces a local feasibility condition that preserves consistency with valid NP verification paths across all possible certificates, ensuring that the maintained computation graph remains feasible at every stage. The total number of extension steps is polynomially bounded in the input size, and each step can be verified in polynomial time.
As a result, the entire graph construction process runs in deterministic polynomial time and decides NP problems without enumerating certificates. This provides a direct and constructive resolution of the P versus NP question. Our result has significant implications for cryptography, combinatorial optimization, and artificial intelligence, where NP-complete problems play a central role.