Actually this class contain problems that admit a polynomial-time r-approximation algorithm for any r > 1.
In this case we have a polynomial time approximation algorithm for any part, so we can decide how must be our solution.
Of course the time must polynomial in the length of the instance, nut we do not require that it is polynomial in the error.
We have an example of a problem in this class, which is minimum partition, for which the time complexity is polynomial both in the length of the instance and in the error we have obtained
In fact, we have shown that the minimum partition by dynamic programming can be shown to be in PTAS.
