An Efficient Algorithmic 3-SAT Formulation for Sudoku Puzzle using Graph Coloring

Abstract

Graph k-Coloring Problem (GKCP) is a renowned NP Complete Problem (NPC) that has been received noteworthy contribution in diverse research areas. One of the important applications of GKCP is the Sudoku puzzle which is also an NPC. We encoded the problem of solving Sudoku puzzle of size (n$\times$n) into GKCP firstly. Further, we reduced GKCP into 3-SAT clauses to obtain the solution of Sudoku puzzle (n$\times$ n). Encoding of Sudoku puzzle to 3-SAT clauses straightforwardly leads to large number of clauses. In this way, we developed an algorithm ${SP}_{2}\mathrm{G}_{2}3$ SAT on the basis of 3-SAT formulation of Sudoku puzzle using GKCP. This algorithm generates $[(n^{2}/2)^{\star}(3n^{2}+(1-2\sqrt{n})^{\star}n-4)+m]3$-SAT clauses that can be solved by SAT solver to obtain the solution of Sudoku puzzle $(n\times n)$. 3-SAT reduction of Sudoku puzzle using GKCP provides an efficient way to acquire the solution of Sudoku puzzle of size $(n\times n)$ as it generates fewer clauses than earlier approach.