# Clarification on NP-hardness and hardness of approximation results for set cover?

I'm not familiar with complexity theory at all so please correct me if I make any incorrect statements.

I am wondering what is the hard case of set cover? My understanding of NP-hardness is that it describes a worse case scenario. In other words, if I am only considering set cover for say the case that the number of subsets is less than the number of elements in the ground set (or vice versa), can I say that this case falls under a 'hard' or 'easy' case of set cover.

Moreover, in looking at the work by Dinur and Steurer 2013 , they say it is NP-hard to approximate set cover within a factor of $$(1−\epsilon)ln (n)$$, where $$n$$ is the size of the universe, but they don't mention the size of the collection of subsets (denote $$m$$), does this imply that this inapproximability result should hold for any $$m$$, regardless of it $$m < n$$ or $$m > n$$?