Sparse Approximation in Lattices and Semigroups

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).

💡 Research Summary

The paper investigates a fundamental sparsity–approximation trade‑off for integer linear systems. Given a full‑row‑rank integer matrix (A\in\mathbb Z^{m\times n}) and a target vector (b) that admits an integer (or non‑negative integer) solution (x) with exactly (n) non‑zero components, the authors ask: for a prescribed sparsity budget (k<n), how well can one approximate (b) by a vector of the form (Ay) where (y) is an integer (or non‑negative integer) vector with at most (k) non‑zero entries?

To formalise the question they introduce the approximation quality function
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