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?