The title pretty much says it all. I want to know if Knapsack is still NP-complete when the weights, values, and weight limit are restricted to a finite set of values. I figure that it wouldn't be because clearly for very small finite sets it isn't. (E.g., if you must use integers from 1 to 3.) That said, I'm struggling to find a polynomial-time algorithm.
One thought I had was since there are a finite set of possible inputs and outputs, could your algorithm just be a massive chain of if statements? I'm not sure whether this even would count as an "algorithm" though.