The knapsack problem, though NP-Hard, is one of a collection of algorithms that can still be approximated to any specified degree.This means that the problem has a polynomial time approximation scheme.
This restriction then means that an algorithm can find a solution in polynomial time that is correct within a factor of (1-ε) of the optimal solution.
There are many variations of the knapsack problem that have arisen from the vast number of applications of the basic problem.
A similar dynamic programming solution for the 0/1 knapsack problem also runs in pseudo-polynomial time. From Definition A, we can know that there is no need for computing all the weights when the number of items and the items themselves that we chose are fixed.
That is to say, the program above computes more than expected because that the weight changes from 0 to W all the time.
The problem often arises in resource allocation where there are financial constraints and is studied in fields such as combinatorics, computer science, complexity theory, cryptography, applied mathematics, and daily fantasy sports.