Sudoku as a SAT Problem

Sudoku is a very simple and well-known puzzle that has achieved international popularity in the recent past. This paper addresses the problem of encoding Sudoku puzzles into conjunctive normal form (CNF), and subsequently solving them using polynomial-time propositional satisfiability (SAT) inference techniques. We introduce two straightforward SAT encodings for Sudoku: the minimal encoding and the extended encoding. The minimal encoding suffices to characterize Sudoku puzzles, whereas the extended encoding adds redundant clauses to the minimal encoding. Experimental results demonstrate that, for thousands of very hard puzzles, inference techniques struggle to solve these puzzles when using the minimal encoding. However, using the extended encoding, unit propagation is able to solve about half of our set of puzzles. Nonetheless, for some puzzles more sophisticated inference techniques are required.