Minimum relevant variables in linear system - additive approximation
Unless $P=NP$, there does not exist a polynomial time approximation algorithm that returns solutions of value at most $\text{OPT}+d$.

Suppose otherwise, and consider some fixed $d$ for which there does exist such an  approximation algorithm $A$. We construct a new multiplicative polynomial time approximation algorithm from $A$, that always returns solutions of value at most $2\cdot\text{OPT}$. Hence consider an arbitrary instance $I$ of Min-RVLS:


For all $k=1,\ldots,d$ check whether $I$ has a solution $x$ with $k$ nonzero variables. This can be done by checking every subset $S$ of $k$ Variables: Set all variables outside $S$ to zero, and solve the linear equation system for the remaining variables. Since there are only $O(n^d)$ subsets to be checked, the overall running time is polynomial. (Here we use that $d$ is a fixed integer.)
If the first step does not reveal any feasible solution, we know that $\text{OPT}&gt;d$. We call the additive approximation algorithm $A$ and find in polynomial time a feasible solution of value at most $\text{OPT}+d&lt;2\cdot\text{OPT}$.


As we either detect an optimal solution (first step) or a solutions with value at most twice the optimum (second step), the resulting polynomial time approximation algorithm
has a multiplicative approximation guarantee of $2$.