Is there any appproximate algorithms in set cover when there are too many sets, say, 2^n sets?

Question

I am recently working on a problem which I think is a fork of the set cover problem. However, the number of sets in my problem is as large as 2^n. And the approximate alogrithms I've found seem to be only effective when there are not too many sets. I wonder there exists an alogorithm that suits with 2^n sets?

Thank you for your answering!!!

Answer 1

No there isn't better approximation than logarithmic one. see wiki:

Inapproximability results show that the greedy algorithm is essentially the best-possible polynomial time approximation algorithm for set cover, under plausible complexity assumptions. Lund & Yannakakis (1994) showed that set covering cannot be approximated in polynomial time to within a factor of 0.72 ln(n), unless NP has quasi-polynomial time algorithms.