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I am a bit confused with the notation of an optimization problem not being approximable within a factor of $(1+n^{\epsilon})$.

What exactly does this mean?

I am confused because if I (as a user of the algorithm) can choose the $\epsilon$ then I can approximate it as good as I want (given the n).

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Most likely it means that there is no algorithm (drawn from some class of "nicely behaved" algorithms, which may be further specified nearby) such that this algorithm always finds a solution within a factor of $(1+n^\epsilon)$ of the true optimum, for an $\epsilon$ that cannot depend on $n$.

It is not quite clear from the wording you quote whether it means that there is no $\epsilon$ with a matching algorithm, or merely that there is some $\epsilon$ for which there is no algorithm.

Even if you had a "nice" algorithm for each $\epsilon$ (perhaps taking $\epsilon$ as an input) such that you can tune the precision as you want, this wouldn't necessarily mean that you could approximate to a fixed precision for all $n$ with "nice" behavior. Is often the case that the complexity analysis that allowed you to think of the algorithm as nicely behaved assumed that $\epsilon$ was constant. If you vary the $\epsilon$, the resource use of the algorithm may grow dramatically in ways that its ordinary big-O asymptotics for constant $\epsilon$ does not reveal.

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