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 ℓ 0 -norm of the vector. Our main results are new improved bounds on the minimal ℓ 0 -norm of solutions to systems A x = b , where A ∈ Z m × n , b ∈ Z m and 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 ℓ 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 R , to other subdomains such as 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. [ABSTRACT FROM AUTHOR]