I am trying to prove whether the following problem is NP-hard or not:
Items with a certain length arrive in a fixed sequence and must be assigned to one of two containers which are constrained in terms of length. The containers are not identical but have different length constraints. Our objective is to maximize the total number of items in both containers. This problem is not really a knapsack problem, because we are not allowed to skip items. The problem is also not exactly a scheduling problem (in which the containers would be two parallel machines and we are assigning a fixed queue of jobs to machines) because we do not want to minimize completion time but want to maximize the number of completed jobs.
I have done some extensive literature review of "Precedence constrained knapsack problems" and "Complexity of Scheduling under Precedence Constraints" but could not yet find this exact type of problem. Most similar problems are indeed NP-hard, so I think this one is as well.
I would appreciate any hints to other literature or tips on how I could prove NP-hardness (or not), e.g. reduction from partitioning / ...
Decision Bin Packing with two bins reduces to your problem. Just take as a sequence any ordering of the objects. The bin packing problem will have a solution if and only if your optimal solution contains the full sequence. Decision Bin Packing is NP-complete with any number of bins, so your problem is NP-hard.
In turn, your problem reduces to Decision Bin Packing: Solve the bin packing problem with the first object of the sequence, then with the two first... until you can't find a solution. This shows that your problem has a weakly polynomial algorithm since Decision Bin Packing with a fixed number of bins is weakly polynomial.