I was reading these notes on approximation algorithms and I stumbled across a PTAS for an optimization version of the partition problem (essentially the Identical-machines scheduling $P_2 \|C_{\operatorname{max}}$ problem). It takes $O(2^{1/\varepsilon})$ time, which is very slow in terms of $\frac{1}{\varepsilon}$. I'm aware of a FPTAS for this problem based on the exact dynamic programming algorithm. But do we really need them in practice? Because on the other hand we have the Longest-processing-time-first scheduling greedy algorithm, which is very fast (takes $O(n \log(n))$ time where $n$ is the number of jobs), very simple and has the approximation ratio $1 + \frac{1}{2L}$ where $L$ is the number of inputs in the greedy part with the max-sum (Wikipedia cites this article for this result). So when the number of jobs $n$ is big enough to block us from using the exact algorithm and force us to resort for approximations, $L$ should be quite big too, and so the LPT algorithm should provide us with an almost optimal solution.
I'm asking the question because I also have in mind the 0-1 Knapsack problem, where, as far as I understand, the most obvious greedy algorithm has the approximation ratio of $2$ and this ratio does not depend on the length of the input: one can construct a tight example for any number of items $n$. So the FPTAS here is of practical usage. Am I wrong?
