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

TL;DR

This paper improves the efficiency of FPTAS algorithms for subset selection problems, especially the Minimum Knapsack Problem, by reducing the running time from quadratic to cubic in the number of items.

Contribution

It introduces a method to achieve faster FPTAS for subset selection problems using existing approximation algorithms, specifically improving the Minimum Knapsack Problem's running time.

Findings

FPTAS for subset selection problems can be optimized using a $ ho$-approximation algorithm.

The new approach reduces the Minimum Knapsack Problem's FPTAS running time from $O(n^5/\epsilon)$ to $O(n^3/\epsilon)$.

The method achieves matching running times for maximization and minimization problems.

Abstract

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.