I’m interested in a variant of minimum set cover where some sets are “incompatible” (they can’t be chosen simultaneously).

To state it more formally:

We have a finite base set $X$ and a family $mathcal{R}$ of subsets of $X$. We also have an undirected graph $G$ with vertex set $mathcal{R}$ and edges representing incompatibilities between sets. The goal is to find a set minimum cover of $X$ which is also an independent set of $G$.

* Does this problem have a name? Or is it a special case of a studied problem? Or can it be reduced to a well-studied problem with a small blowup in runtime?* (by “small blowup” I mean not simply polynomial, but preferably a small degree polynomial).

I’m particularly interested in the geometric variant where $X$ is a set of points and the sets in $mathcal{R}$ are defined as the intersection of $X$ with some simple geometric ranges (in which case tools such as VC-dimension, $epsilon$-nets, range-searching and so on might come in handy for approximation algorithms). But I welcome any relevant reference.