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: Ax = b, where A is of size m times n and n > m. We then prove that the Sudoku solution is the sparsest solution of Ax = b, which can be obtained by lo norm minimization, i.e. min ||x:||₀ s.t. Ax = b. Instead of this minimization SB problem, inspired by the sparse representation literature, we solve the much simpler linear programming problem of minimizing the l₁ norm of x, i.e. min ||x||₁ s.t. Ax = b, and show numerically that this approach solves representative Sudoku puzzles.