Motivated by problems in optimization we study the sparsity of the solutions to systems of linear Diophantine equations and linear integer programs, i.e., the number of non-zero entries of a solution, which is often referred to as the $\ell_0$-norm. Our main results are improved bounds on the $\ell_0$-norm of sparse solutions to systems $A x = b$, where $A \in \mathbb{ Z}^{ m \times n}$, $b \in \mathbb{ Z}^m$ and $x$ is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In the lattice case and certain scenarios of the semigroup case, we give polynomial time algorithms for computing solutions with $\ell_0$-norm satisfying the obtained bounds.
Categories: math.OC