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).

翻译：本文探讨以下问题：假设系统 $Ax = b$ 存在整数解或非负整数解 $x$，且 $x$ 的非零分量个数为 $n$。对于给定的自然数 $k < n$，目标是利用 $Ay$ 逼近 $b$，其中 $y$ 是至多包含 $k$ 个非零分量的整数解或非负整数解。我们建立了该问题的一般性上界。在特定情形下，这些上界是紧的。若将逼近质量视为参数 $k$ 的函数，本文解释了为何当 $k$ 趋近于 $n$ 时，逼近质量呈指数级提升。本文是发表于第26届整数规划与组合优化国际会议（IPCO）的扩展摘要的完整版本。