Construction of Heuristics for a Search-Based Approach to Solving Sudoku

Abstract

Sudoku is a logic puzzle, consisting of a 9×9 grid and further subdivided into ‘mini-grids’ of size 3×3. Each row, column, and 3×3 mini-grid contains the numbers 1 to 9 once, with a true Sudoku grid having a unique solution. Sudoku, along with similar combinatorial structures, has relationships with a range of real-world problems. Much published work on the solution of Sudoku puzzles has acknowledged the link between Sudoku and Latin Squares, thereby recognising the scale of any search space of possible solutions and that the generalization of the puzzle to larger grid sizes is NPcomplete. However, most published approaches to the solution of Sudoku puzzles have focussed on the use of constraint satisfaction algorithms that effectively mimic solution by hand, rather than directly exploiting features of the problem domain to reduce the size of the search space and constructing appropriate heuristics for the application of search techniques. This paper highlights important features of the search space to arrive at heuristics employed in a modified steepest ascent hill-climbing algorithm, and proposes a problem initialization and neighbourhood that greatly speed solution through a reduction of problem search space. Results shown demonstrate that this approach is sufficient to solve even the most complex rated puzzles, requiring relatively few moves. An analysis of the nature of the problem search space is offered.