A note on "Approximation schemes for a subclass of subset selection problems", and a faster FPTAS for the Minimum Knapsack Problem

Pruhs and Woeginger prove the existence of FPTAS's for a general class of minimization and maximization subset selection problems. Without losing generality from the original framework, we prove how better asymptotic worst-case running times can be achieved if a $\rho$-approximation algorithm is available, and in particular we obtain matching running times between maximization and minimization subset selection problems. We directly apply this result to the Minimum Knapsack Problem, for which the original framework yields an FPTAS with running time $O(n^5/\epsilon)$, where $\epsilon$ is the required accuracy and $n$ is the number of items, and obtain an FPTAS with running time $O(n^3/\epsilon)$, thus improving the running time by a quadratic factor in the worst case.