Approximating the 3-period Incremental Knapsack Problem

Abstract

The 0–1 Incremental Knapsack Problem (IKP) is a generalization of the standard 0–1 Knapsack Problem (KP) 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 problem calls for maximizing the sum of the profits over the whole time horizon. In this work, we consider the case with three time periods where we assume that each item can be packed also in the first time period. We propose an approximation algorithm with a tight approximation ratio of 3037. We strongly rely on Linear Programming (LP) to derive this bound showing how the proposed LP-based analysis can be seen as a valid alternative to more formal proof systems.