I'm starting to dabble in the world of approximation algorithms and had a question about the convention many papers will use when talking about the approximation factor. I know that an approximation factor will give the factor within which the algorithm will perform with respect to the optimal solution. This is pretty easy to grasp for things like a 2-factor approximation. I've also seen other literature and lecture notes refer to $1-1/\epsilon$ when giving the probability and/or percentage of the optimal that will be returned (or, put another way, how "much" of the optimal an approximation algorithm will, well, approximate).
But I've seen some papers (like Linear-Time Approximation for Maximum Weight Matching ) refer to a $1-\epsilon$ approximation ratio. For example:
we give an algorithm that computes a (1 − ε)-approximate maximum weight matching
So my question is: are they referring to the $1-1/\epsilon$ ratio that I've been seeing elsewhere? Apologies if this is a simplistic question, I'm just getting starting in learning about these very-cool algorithms and am trying to get my head around which papers are presenting things that are extraordinary vs. not as competitive compared to the optimal solution. Thank you!