I read it in not only one place. People write theorems of the form:
Theorem: ALG is an O(1)-approximation algorithm
It means that ALG is a constant factor approximation algorithm but is it safe to say that? I mean it misleads. Why am I asking?
Because, many problems I know of do not have an $\alpha$-approximation algorithm where $\alpha<\gamma$, unless P=NP. For example, the k-center problem does not have an $\alpha$-approximation algorithm where $\alpha<2$, unless P=NP. Well, there is a simple greedy algorithm that is 2-approximation algorithm for the k-center problem. Can we say that this greedy is O(1)-approximation algorithm for the k-center problem? I mean, if it is an O(1)-approximation algorithm for the k-center problem, it is, for example, an 1.9-approximation algorithm, no?