Approximation Results for the Incremental Knapsack Problem

Abstract

We consider the 0–1 Incremental Knapsack Problem where the knapsack capacity grows over time periods and if an item is placed in the knapsack in a certain period, it cannot be removed afterwards. The problem calls for maximizing the sum of the profits over the whole time horizon. In this work, we manage to prove the tightness of some approximation ratios of a general purpose algorithm currently available in the literature. We also devise a Polynomial Time Approximation Scheme (PTAS) when the input value indicating the number of periods is considered as a constant. Then, we add the mild and natural assumption that each item can be packed in the first time period. For this variant, we discuss different approximation algorithms suited for any number of time periods and provide an algorithm with a constant approximation factor of $\frac{6}{7}$ for the case with two periods.