Uniqueness of integer solution of linear equations

We consider the system of m linear equations in n integer variables Ax = d and give sufficient conditions for the uniqueness of its integer solution x ∈ {−1,1}n by reformulating the problem as a linear program. Uniqueness characterizations of ordinary linear programming solutions are utilized to obtain sufficient uniqueness conditions such as the intersection of the kernel of A and the dual cone of a diagonal matrix of ±1’s is the origin in Rn. This generalizes the well known condition that ker(A) = 0 for the uniqueness of a non-integer solution x of Ax = d. A zero maximum of a single linear program ensures the uniqueness of a given integer solution of a linear equation.