MHB Versions of the Knapsack problem

[mathmari]

Hey!


I am asked to state the two versions of the Knapsack problem and their differences. Could I formulate it as followed?? (Wondering)

The Knapsack problem is the following:

There are $n$ items, where the $i^{th}$ item has a benefit of $v_i$ and it has weight $w_i$.

We want to pick some items so that we maximize the total benefit while keeping the total weight of $W$.

The difference between the integer and the fractional version of the Knapsack problem is the following:

At the integer version we want to pick each item either fully or we don't pick it.

At the fractional version we can take a part of the item.

 

[mathmari]

With what type of programming can each of the two versions be solved?

Which of the two versions has the optimal substructure property ?

Which of the two versions has the greedy choice property ?
For the greedy choice property I thought the following:

Wir choose at each step the "best" item, which is the one with the greatest benefit and the smallest weight. We do the same choice until the Knapsack is full or there aren't any objects that fill in the Knapsack.

Is this correct? The optimal substructure property means that each subproblem we have the optimal solution.

Right? But how can we check which of the versions has these properties?