Given an integer or a non-negative integer solution $$\boldsymbol{x}$$ to a system $$\boldsymbol{A}\boldsymbol{x}= \boldsymbol{b}$$ , where the number of non-zero components of $$\boldsymbol{x}$$ is at most n. This paper addresses the following question: How closely can we approximate $$\boldsymbol{b}$$ with $$\boldsymbol{A}\boldsymbol{y}$$ , where $$\boldsymbol{y}$$ is an integer or non-negative integer solution constrained to have at most k non-zero components with $$k