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.