# A variant of hitting set problem? Is this also a NP-hard problem?

Let’s start from finding a minimum hitting set problem. Given a collection of sets $$U=\{\{S_1\},\{S_2\},\{S_3\},\{S_4\},\{S_5\},\{S_6\}\}=\{\{1, 2, 3\}, \{1, 3, 4\}, \{1, 4, 5\}, \{1, 2, 5\}, \{2, 3\}, \{4, 5\}\}$$,
it is easy to know that a minimum hitting set is $$\{2,4\}$$.

I am thinking what this problem would be if the set $$S$$ is also a collection of sets. For instance, given $$S_1=\{\{1,2,3\},\{3,4\},\{1,2\}\}$$,
$$S_2=\{\{3\},\{3,5\},\{1,3\}\}$$,
$$S_3=\{\{2,5\},\{4\},\{1,5\},\{1,10,6,7\}\}$$,
then a minimum “hitting set” is $$\{3,4\}$$. That is because $$\{3,4\}$$ only consists of two elements and hits every set (i.e., $$\{3,4\}\in S_1$$, $$\{3\}\subset \{3,4\}$$ and $$\{4\}\subset \{3,4\}$$).

Does anyone have an idea to solve this problem? Do you think it is a variant of hitting set problem? Is it an NP-hard problem?