Linear Systems, Sparse Solutions, and Sudoku

In this paper, we show that Sudoku puzzles can be formulated and solved as a sparse linear system of equations. We begin by showing that the Sudoku ruleset can be expressed as an underdetermined linear system: ${mmb{Ax}}={mmb b}$, where ${mmb A}$ is of size $mtimes n$ and $n>m$. We then prove that the Sudoku solution is the sparsest solution of ${mmb{Ax}}={mmb b}$, which can be obtained by $l_{0}$ norm minimization, i.e. $minlimits_{mmb x}Vert{mmb x}Vert_{0}$ s.t. ${mmb{Ax}}={mmb b}$. Instead of this minimization problem, inspired by the sparse representation literature, we solve the much simpler linear programming problem of minimizing the $l_{1}$ norm of ${mmb x}$, i.e. $minlimits_{mmb x}Vert{mmb x}Vert_{1}$ s.t. ${mmb{Ax}}={mmb b}$, and show numerically that this approach solves representative Sudoku puzzles.