Sparse representation of vectors in lattices and semigroups

Iskander Aliev, Gennadiy Averkov, Jesús A. De Loera, Timm Oertel

Abstract: 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.