Is this a known problem?

Given a set of sets $S$ find a set of sets $B$ s.t. each set in $S$ can be obtained through unions of some sets in $B$. The set $S$ is already a solution but the objective is to find the smallest $B$.

For instance $\{\{a,b\}, \{c\}, \{a,b,c\}\}$ can be obtained from $\{\{a,b\}, \{c\}\}$.

Since it's somewhat similar to set-cover, I suspect it's NP-hard.

Additional remark: $B$ does not have to be a subset of $S$.


1 Answer 1


A decision variant, without the minimality condition, asking whether there is a set $B$ of size $n$ is called the set basis problem [SP7] in Computers and Intractability: A Guide to the Theory of NP-Completeness by Garey and Johnson. The NP-completeness of it was proved by Stockmeyer.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.