A Faster FPTAS for the Unbounded Knapsack Problem

TL;DR

This paper introduces an improved Fully Polynomial Time Approximation Scheme (FPTAS) for the Unbounded Knapsack Problem, achieving faster running time and reduced space complexity compared to the longstanding best algorithm, with implications for related packing problems.

Contribution

The authors develop a new FPTAS for UKP with significantly improved running time and space bounds over the previous best, enhancing efficiency for approximation algorithms.

Findings

FPTAS running time improved to O(n + (1/ε)^2 log^3(1/ε))

Space complexity reduced to O(n + (1/ε) log^2(1/ε))

Impacts approximation schemes for Bin Packing and Strip Packing

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.