A Faster FPTAS for the Unbounded Knapsack Problem

Abstract

The Unbounded Knapsack Problem (UKP) is a well-known variant of the famous 0-1 Knapsack Problem (0-1 KP). In contrast to 0-1 KP, an arbitrary number of copies of every item can be taken in UKP. Since UKP is NP-hard, fully polynomial time approximation schemes (FPTAS) are of great interest. Such algorithms find a solution arbitrarily close to the optimum $\mathrm {OPT}(I)$, i.e. of value at least $(1-\varepsilon ) \mathrm {OPT}(I)$ for $\varepsilon > 0$, and have a running time polynomial in the input length and $\frac{1}{\varepsilon }$. For over thirty years, the best FPTAS was due to Lawler with a running time in $O(n + \frac{1}{\varepsilon ^3})$ and a space complexity in $O(n + \frac{1}{\varepsilon ^2})$, where n is the number of knapsack items. We present an improved FPTAS with a running time in $O(n + \frac{1}{\varepsilon ^2} \log ^3 \frac{1}{\varepsilon })$ and a space bound in $O(n + \frac{1}{\varepsilon } \log ^2 \frac{1}{\varepsilon })$. This directly improves the running time of the fastest known approximation schemes for Bin Packing and Strip Packing, which have to approximately solve UKP instances as subproblems.

Research supported by DFG project JA612/14-2, “Entwicklung und Analyse von effizienten polynomiellen Approximationsschemata für Scheduling- und verwandte Optimierungsprobleme”.