Polynomial Time Algorithm for Solving Sudoku Problems

The NP=P? (NP equals P) problem, as one of the seven Millennium Mathematics Problems, continues to challenge researchers in mathematics and computer science. The core of this question lies in exploring whether the complexity classes P and NP are equivalent. NP-complete problems, as the most challenging subset within NP, would prove P=NP if any NP-complete problem could be solved in polynomial time. Sudoku solving, a typical NP-complete problem, has yet to find a polynomial-time solution algorithm. This paper proposes a polynomial-time algorithm for solving Sudoku, aiming to demonstrate that NP-complete problems can be solved in polynomial time, thereby proving P=NP. The algorithm not only applies to standard 9x9 Sudoku grids but can also be extended to Sudoku grids of arbitrary size, significantly reducing computational complexity compared to non-polynomial-time algorithms. This research holds important theoretical significance for exploring the NP=P? problem.