Sparse representation of vectors in lattices and semigroups

We study the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the $$\ell _0$$ -norm of the vector. Our main results are new improved bounds on the minimal $$\ell _0$$ -norm of solutions to systems $$A\varvec{x}=\varvec{b}$$ , where $$A\in \mathbb {Z}^{m\times n}$$ , $${\varvec{b}}\in \mathbb {Z}^m$$ and $$\varvec{x}$$ is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In certain cases, we give polynomial time algorithms for computing solutions with $$\ell _0$$ -norm satisfying the obtained bounds. We show that our bounds are tight. Our bounds can be seen as functions naturally generalizing the rank of a matrix over $$\mathbb {R}$$ , to other subdomains such as $$\mathbb {Z}$$ . We show that these new rank-like functions are all NP-hard to compute in general, but polynomial-time computable for fixed number of variables.