A Polynomial Time Algorithm for Solving Sudoku Problems

The P vs. NP equivalence problem, one of the seven major mathematical challenges of the millennium, continues to challenge researchers in mathematics and computer science. At its core, this question explores whether the complexity classes P and NP are equivalent. NPC problems, the most challenging subset of NP problems, can lead to the proof that P=NP if a polynomial-time algorithm is found for any of them. The Sudoku solving problem, a typical NPC problem, has yet to be solved using a polynomial-time algorithm. This paper introduces a polynomial-time algorithm for solving Sudoku, aiming to demonstrate that NPC problems have polynomial-time solutions, thereby proving P=NP. This algorithm is not only applicable to standard 9x9 Sudoku but can also be extended to any grid size, significantly reducing computational complexity compared to traditional enumeration methods and their derivatives. The findings of this study may hold significant theoretical implications for exploring the P=NP problem.