# Can PTAS be used to optimally solve Knapsack?

Suppose you have a Knapsack (optimisation) problem with integer values and weights, and you know the optimal value $$OPT$$. Can you compute an optimal solution in polynomial time by using a PTAS or FPTAS and setting $$\epsilon < 1/OPT$$, since this would mean it computes a solution in polynomial time which is of value $$\geq (1-\epsilon) OPT > OPT - 1$$ which due to values being integers, means it must be an optimal solution?

Also, does this idea generalise by setting $$\epsilon$$ to something so that $$\epsilon OPT$$ is smaller than the smallest difference in value between items? This can even be done without knowledge of $$OPT$$ by searching for the smallest difference $$d$$ in time $$O(n^2)$$ for $$n$$ items and then setting $$\epsilon < d / V$$ with $$V$$ being the sum of all item values.

Is there perhaps a catch about Knapsack being a weakly NP-hard problem, meaning $$OPT$$ isn't a value bounded polynomially by the input size (even if $$OPT$$ is given, since it's given in binary) and thus the runtime of the PTAS or FPTAS isn't polynomial...?

The runtime of a FPTAS is polynomial in the input size and in $$1/\epsilon$$. However, if $$\epsilon$$ depends on parameters which may not be polynomial in the input size, such as in this case $$OPT$$ or $$V$$, then the algorithm is not a polynomial time algorithm.