Optimizing Sparsity over Lattices and Semigroups

Abstract

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 [Formula Presented]-norm. Our main results are improved bounds on the [Formula Presented]-norm of sparse solutions to systems [Formula Presented], where [Formula Presented], [Formula Presented] and [Formula Presented] 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 [Formula Presented]-norm satisfying the obtained bounds.