Solving Sudoku with Ant Colony Optimisation

Sudoku is a well-known logic-based puzzle game that was first published in 1979 under the name of "Number Place". It was popularised in Japan in 1984 by the puzzle company Nikoli, and later named "Sudoku", which roughly translates to "single digits". The puzzle gained attention in the West in 2004, after The Times published its first Sudoku grid (at the instigation of Hong Kong-based judge Wayne Gould, who first encountered the puzzle in 1997, and developed a computer program to automatically generate instances). Sudoku is now a global phenomenon, and many newspapers now carry it alongside their existing crosswords (see [4] for a general history of the puzzle). The simplest variant of Sudoku uses a 9 9 grid of cells divided into nine 3 3 subgrids (Figure 1 (left)). The aim of the puzzle is to fill the grid with digits such that each row, each column, and each 3 3 subgrid contains all of the digits 1-9 (Figure 1 (right)). An instance of Sudoku provides, at the outset, a partially-completed grid, but the difficulty of any grid derives more from the range of techniques required to solve it than the number of cell values that are provided for the player. Sudoku is an NPcomplete problem [12], as first shown in [35] (via a reduction from the Latin Square Completion problem [2]).