set cover where only certain special subsets are allowed

Question

I am trying to solve a problem which turns out to be a form of the set cover problem. I've implemented the greedy Set cover approximation algorithm for set cover, but it turns out to not be accurate enough for my needs.

Specifically I have a fixed, finite, list of special sets and I am trying to create a function which when given a set returns the smallest list of special sets which cover that given set, and that list should contain no sets which contain elements not in that given set. The fixed list seems like it should make the problem easier to solve than set cover, but I'm not sure how the additional requirement of no extra elements affects the difficulty. My sets also have a known maximum size which should also make this easier to solve.

A toy version of the problem is as follows:

Let $U$ be the set $\{a,b,c,d,e,f,g,h\}$. All the sets we mention from here on will be subsets of this set.

Let $S$ be a list of special sets. For this version of the problem we can use the following list:

$$ S = [\{a,b,c,d,e,f,g,h\}, \{a,b,c,d\}, \{e,f,g,h\}, \{a,e\}, \{b,f\}, \{c,g\}, \{d,h\}, \{a,b\}, \{b,c\}, \{c,d\}, \{d,e\}, \{e,f\}, \{f,g\},\{g,h\}, \{a\}, \{b\}, \{c\}, \{d\}, \{e\}, \{f\}, \{g\}, \{h\}]$$

For a given set $G$ what is the smallest list of sets which are part of $S$ which cover $G$, (that is where the union of the sets in the list is $G$).

For the above toy problem, the greedy approximation I've implemented, if presented with a set like $\{b, c, g, h\}$, produces $[\{c, g\}, \{b\}, \{h\}]$ instead of the smaller list $[\{b, c\}, \{g, h\}]$.

Has this sort of problem been well studied? Is there a known non-approximate algorithm given the additional constraints, that would be better than naive dynamic programming? A well-known name for the version of set cover where extra elements are not allowed would also be useful.