Optimizing sparsity over lattices and semigroups

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 ℓ0-norm. Our main results are improved bounds on the ℓ0-norm of sparse solutions to systems Ax=b, where A∈Zm×n, b∈Zm 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 ℓ0-norm satisfying the obtained bounds