Limitations of Incremental Dynamic Programs

We consider so-called “incremental” dynamic programming algorithms, and are interested in the number of subproblems produced by them. The standard dynamic programming algorithm for the n-dimensional Knapsack problem is incremental, produces nK subproblems and nK relations (wires) between the subproblems, where K is the capacity of the knapsack. We show that any incremental algorithm for this problem must produce about nK subproblems, and that about nK logK wires (relations between subproblems) are necessary.