  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.
