# Maximization problem on finite collection of finite sets

## Problem

I am considering the following maximization problem:

• Input is a finite collection of finite sets $$\mathcal{F} = \{ X_1, X_2, \ldots, X_n \}$$.
• Goal is to find a subset $$G \subseteq \mathcal{F}$$ that maximizes $$|G| \times |\bigcap G|$$ where
• $$|G|$$ is the cardinality of the set $$G$$, and
• $$\bigcap G = \bigcap \{X_{i_1}, X_{i_2}, \ldots, X_{i_m} \}$$.

As an example, for the collection $$\mathcal{F} = \{ \{a, b, c\}, \{a, b, c, x\}, \{b, c, y\}, \{a, b, c, z\} \},$$ the maximizing subset is $$G = \{ \{a, b, c\}, \{a, b, c, x\}, \{a, b, c, z\} \}$$ and the score is $$3 \times |\{a, b, c\}| = 9$$.

Note: the score of $$\mathcal{F}$$ itself is $$4 \times |\{b, c\}| = 8$$.

## Question

I am planning to use a procedure of this problem for compressing data (represented by finite collections of finite sets). However, I don’t have any good idea to solve this problem efficiently. As yow know, we can solve this by enumerating all the collections of $$\mathcal{F}$$; but, it’s too slow for practical use.

Is there a polynomial-time or some kind of efficient algorithm for this problem? Or, does this problem belong to the complexity class that cannot be solved in polynomial time?