Knapsack with continuous (non distinct) constraint

Question

I watched Dynamic Programming - Kapsack Problem (YouTube). However, I am solving a slightly different problem where the constraint is the budget, price, in double, not integer.

Answer 1

Have your looked at this. Sorry, I don't have comment privilege.

Edit 1

Are you saying constraint is the budget instead of knapsack weight? This still remains a knapsack problem.

Or are your saying instead of Item Values as Integers(0-1 knapsack problem) your have fractions. Then Greedy approach should do fine.

Edit 2

If I understand your problem correctly.. It states

We have n kinds of items, 1 through n. Each kind of item i has a value vi and a price pi. We usually assume that all values and pricess are nonnegative. The Budget is B.

The most common formulation of the problem is the 0-1 knapsack problem, which restricts the number xi of copies of each kind of item to zero or one. Mathematically the 0-1-knapsack problem can be formulated as:

         n
maximize E(vi.xi)
         i=i

           n
subject to E(pi.xi) <= B,         xi is a subset of {0,1}
           i=1

Neo Adonis's answer is spot on here.. Dynamic programming wont work for arbitrary precision in practice.

But if you are willing to limit the precision say to 2 decimal places.. then carry on as explained in video.. your table should look something like this..

+------+--------+--------+--------+--------+--------+--------+
| Vi,Pi| 0.00   | 0.01   | 0.02   | 0.03   | 0.04 ...    B   |
+------+--------+--------+--------+--------+--------+--------+
|4,0.23|        |        |        |        |        |        |
|2,2.93|        |        |        |        |        |        |
|7,9.11|        |        |        |        |        |        |
| ...  |        |        |        |        |        |        |
| Vn,Pn|        |        |        |        |        | answer |
+------+--------+--------+--------+--------+--------+--------+

you can even convert real numbers to int as you have mentioned.

Yes, the range of values is very large, and you also have to understand knapsack is NP-complete, i.e, there is no efficient algorithm to solve this. only pseudo polynomial solution using DP. see this and this.