Theorem: Unless $NP \subset DTIME (n^{O(\log \log n)})$, there is no $(1-o(1))\ln n$-approximation for set cover problem.
I am a bit confused by this theorem. As we know, greedy algorithm is $(\ln n+1)$-approximation, does this mean greedy algorithm is almost the best algorithm for set cover problem?
In the wiki set cover problem, there is a very bad example about the greedy algorithm, so I think a $\ln n$-approximation is meaningless. Does the theorem above say that it is impossible to design a constant approximation algorithm?