# PTAS vs. FPTAS input

I am trying to understand what is the PTAS, FPTAS and what is the difference between them.

I found this analysis:

PTAS definition vs. FPTAS

but I cannot understand what do we mean by saying:

"....time complexity is polynomial in the input size and also polynomial in 1/ϵ" ?? Ok n has to do with the linear O(n) worst case that the algorithm runs. But O(1/ε) how does it work?

For example, the running time $$O(2^{1/\varepsilon} \cdot n^2)$$ corresponds to a PTAS, since the running time depends polynomialy on $$n$$, but not on $$1/\varepsilon$$ since it is in the exponent. The running time $$O(1/\varepsilon \cdot n \log{n})$$ depends polynomialy on $$n$$ and $$1/\varepsilon$$, so it corresponds to a FPTAS.
NB, you also need to show that your algorithm has an approximation ratio of $$(1 + \varepsilon)$$ (or -) to show the algorithm is in fact a (F)PTAS.