Dating problem cryptography
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.
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.
A multiple constrained problem could consider both the weight and volume of the boxes.
Preferably, however, the approximation comes with a guarantee on the difference between the value of the solution found and the value of the optimal solution.
As with many useful but computationally complex algorithms, there has been substantial research on creating and analyzing algorithms that approximate a solution.
To be exact, the knapsack problem has a fully polynomial time approximation scheme (FPTAS)..
It then proceeds to insert them into the sack, starting with as many copies as possible of the first kind of item until there is no longer space in the sack for more.
The knapsack problem, though NP-Hard, is one of a collection of algorithms that can still be approximated to any specified degree.