Question

I was thinking,

I wanted to do a variation on the Knapsack Problem.

Imagine the original problem, with items with various weights/value.

My version will, along with having the normal weights/values, contain a “group” value.

eg.
Item1[5kg, $600, electronic]
Item2[1kg, $50, food]

Now, having a set of items like this, how would I code up the knapsack problem to make sure that a maximum of 1 item from each “group” is selected.

Notes:

1. You don’t need to choose an item from that group
2. There are multiple items in each group
3. You’re still minimizing weight, maximizing value
4. The amount of groups are predefined, along with their values.

I’m just writing a draft of the code out at this stage, and I’ve chosen to use a dynamic approach. I understand the idea behind the dynamic solution for the regular knapsack problem, how do I alter this solution to incorporate these “groups”?

KnapSackVariation(v,w,g,n,W)
{
  for (w = 0 to W)
     V[0,w] = 0;
  for(i = 1 to n)
     for(w = 0 to W)
        if(w[i] <= w)
           V[i,w] = max{V[i-1, w], v[i] + V[i-1, w-w[i]]};
        else
           V[i,w] = V[i-1, w];
     return V[n,W];
}

That’s what I have so far, need to add it so that it will remove all corresponding items from the group it is in each time it solves this.

Answer 1

Assume
c[i] : The category of the ith element
V[i,w,S] : Maximum value of the knapsack such that it contains at max one item from each category in S

Recursive Formulation
V[i,w,S] = max(V[i-1,w,S],V[i,w-w[i],S-{c[i]}] + v[i])
Base Case
V[0,w,S] = -`infinity if w!=0 or S != {}`