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Say we have a collections of sets $\mathcal X = \{X_1, \dots, X_n\}$ (not necessarily disjoint), and we want to count the number of possible unions of sets in $\mathcal X$, i.e. the size of $\{\bigcup_{Y \in \mathcal Y} Y \,|\, \mathcal Y \subseteq \mathcal X \}$.

Is there a name for this problem? What are the fastest algorithms to solve this problem?

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I found the answer here. It is called the union closure and the complexity appears to be the same as computing the number of maximal independent sets in a bipartite graph.

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