Sparse Approximation in Lattices and Semigroups

TL;DR

This paper investigates how well one can approximate solutions to linear systems with sparse integer vectors, establishing bounds and explaining the exponential improvement in approximation quality as sparsity increases.

Contribution

It provides upper bounds for sparse approximation in lattices and semigroups, with tight bounds in specific cases, and explains the exponential growth in approximation quality.

Findings

Upper bounds for sparse approximation are established.

Bounds are tight in specific cases.

Approximation quality improves exponentially with increased sparsity.

Abstract

This paper deals with the following question: Suppose that there exist an integer or a non-negative integer solution $x$ to a system $Ax = b$, where the number of non-zero components of $x$ is $n$. The target is, for a given natural number $k < n$, to approximate $b$ with $Ay$ where $y$ is an integer or non-negative integer solution with at most $k$ non-zero components. We establish upper bounds for this question in general. In specific cases, these bounds are tight. If we view the approximation quality as a function of the parameter $k$, then the paper explains why the quality of the approximation increases exponentially as $k$ goes to $n$. This paper is a complete version of an extended abstract that appeared at the 26th International Conference on Integer Programming and Combinatorial Optimization (IPCO).