Approximating the Incremental Knapsack Problem

TL;DR

This paper studies the 0-1 Incremental Knapsack Problem, providing approximation algorithms, LP-based bounds, and a PTAS, with insights into variants where items can be packed initially or across multiple periods.

Contribution

It introduces new approximation results, including a PTAS for fixed periods and LP-based analysis methods, for the incremental knapsack problem and its variants.

Findings

Proved tight approximation ratios for existing algorithms.

Developed a PTAS for a fixed number of periods.

Analyzed variants with initial packing constraints.

Abstract

We consider the 0-1 Incremental Knapsack Problem (IKP) where the capacity grows over time periods and if an item is placed in the knapsack in a certain period, it cannot be removed afterwards. The contribution of a packed item in each time period depends on its profit as well as on a time factor which reflects the importance of the period in the objective function. The problem calls for maximizing the weighted sum of the profits over the whole time horizon. In this work, we provide approximation results for IKP and its restricted variants. In some results, we rely on Linear Programming (LP) to derive approximation bounds and show how the proposed LP-based analysis can be seen as a valid alternative to more formal proof systems. We first manage to prove the tightness of some approximation ratios of a general purpose algorithm currently available in the literature and originally applied to a time-invariant version of the problem. 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 for the special case with two periods.