Optimizing Sparsity over Lattices and Semigroups

TL;DR

This paper investigates the sparsity of solutions to linear Diophantine systems and integer programs, providing improved bounds on the number of non-zero entries and polynomial algorithms for certain cases.

Contribution

It offers new bounds on the sparsity of solutions and develops polynomial time algorithms for computing solutions with bounded non-zero entries in specific cases.

Findings

Improved bounds on the $oldsymbol{ ext{l}_0}$-norm of solutions.

Polynomial time algorithms for lattice and certain semigroup cases.

Enhanced understanding of sparsity in integer solutions to linear systems.

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 $\ell_0$-norm. Our main results are improved bounds on the $\ell_0$-norm of sparse solutions to systems $A x = b$, where $A \in \mathbb{Z}^{m \times n}$, $b \in \mathbb{Z}^m$ 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 $\ell_0$-norm satisfying the obtained bounds.