Is there any literature about the complexity of the integer knapsack problem with bounded weights? To make it clear, I want an optimal solution to the following problem:$\max \sum_{i=1}^k c_i \cdot x_i$$\sum_{i=1}^k w_i \cdot x_i \leq W$$x_i \in \{0,\ldots,k_i\}$where $k_i$ is an item-specific limit for the number of copies that can be taken of item $i$ and $w_i$ is integral but bounded by $k$.I'm not sure whether it is NP-hard or not...in contrast to the traditional knapsack problem, it would be possible to iterate over $w_i$ here (in polynomial time), but not to iterate over the whole sum since the values $k_i$ are not bounded by a polynomial. On the other hand, I cannot imagine a reduction of an $NP$-hard problem. Any ideas?
Integer knapsack problem with bounded weights
knapsack problems