# Minimal partition covering?

I am working on a problem that arises in the design of experiments. I wonder if it is part of a well-studied class of problems.

The problem is:

Start with a set of points $$S$$ and a target partition of that set $$P$$.

Let $$\mathcal{P}$$ be a set of partitions of $$S$$ not including $$P$$

The problem is to “cover” $$P$$ using the minimal number of partitions from $$\mathcal{P}$$

To make this more formal, a subset of the set of partitions $$\tilde{\mathcal{P}}\subset \mathcal{P}$$ covers $$P$$ if the meet of $$\tilde{\mathcal{P}}$$ (the coarsest partition finer than all partitions in $$\tilde{\mathcal{P}}$$) is finer than $$P$$.

Example

For example, suppose the target partition is $$P=\big\{\{1,2\},\{3\},\{4\},\{5,6\}\big\}$$ and $$\mathcal{P}=\Big\{\big\{\{1\},\{2,3,4,5,6\}\big\};\big\{\{1,2,3\},\{4,5,6\}\big\};\big\{\{1,2\},\{3,4\}\{5,6\}\big\}\Big\}$$

We only need two partitions from $$\mathcal{P}$$ to cover this. The meet of $$\Big\{\big\{\{1,2,3\},\{4,5,6\}\big\};\big\{\{1,2\},\{3,4\}\{5,6\}\big\}\Big\}$$ is $$P$$ and this is the minimal way to cover $$P$$ from $$\mathcal{P}$$.

What I’ve tried:

I have tried to map my problem into a set covering problem with no luck. I got hung up with the meet operation operating differently than the union in set covering.