Before I even attempt coming up with an efficient algorithm, I tried googling for similar problems but didn't get far, most queries mentioning "sets" in them led to some sort of Multiple subset sum or other subset construction problem. What I'm looking for is quite literally the opposite.
The problem is very straightforward:
Given a subset of a power set, I need to tell if I can reconstruct the universal set only by using disjoint subsets.
Example 1:
$x = \{1, 2, 3, 4, 5\}$
$x_s = \{\{1\}, \{2\}, \{2, 3\}, \{3, 4, 5\}\}$
$\longrightarrow$ Reconstruction possible with $\{1\}, \{2\}, \{3, 4, 5\}$.
Example 2:
$x = \{1, 2, 3, 4, 5\}$
$x_s = \{\{1, 2\}, \{2, 3\}, \{3, 4\}, \{4, 5\}, \{1, 3, 5\}, \{4\}\}$
$\longrightarrow$ Reconstruction not possible with sets such that all sets would be pairwise disjoint.
An optimal algorithm does not seem trivial at all, but I also don't believe that I am the first person to think of this type of problem. Are there any resources regarding such problems? Could it be rephrased to become some special case of Knapsack problem?
Also, since I'm only interested in whether it's possible or not to reconstruct the universal set, maybe there's some trick I am missing that could let me avoid going through the complete tree of possible unions?