Producing a Family of 0,1 Knapsack Sets

[bacte2013]

Dear Physics Forum friends,

I am currently stuck with the following question about the integer optimization:

"Produce a family of 0,1 knapsack sets (having an increasing number n of variables) whose associated family of minimal covers grows exponentially with n."

My thought is that I need to partition the set based on taking each element from indices based on the constraint, but I am not sure. Frankly speaking, I am clueless as how to start solving this problem...Could you help me out?

 

[mighty2000]

Thank you for your question. The problem you are facing is a classic one in integer optimization and can be solved using various techniques. One approach to solving this problem is by using dynamic programming.

To start, let's define the knapsack problem and its associated terms. The knapsack problem is a combinatorial optimization problem where given a set of items with a weight and value, the goal is to maximize the value of items that can fit into a knapsack with a given weight capacity. The 0,1 knapsack problem is a special case where the items can only be either fully included or excluded from the knapsack.

Now, to produce a family of knapsack sets with an increasing number of variables, we can start with a basic set of items and gradually add more items with each new variable. For example, let's say our initial set has 3 items with weights and values as follows:

Item 1: weight = 2, value = 10
Item 2: weight = 3, value = 15
Item 3: weight = 4, value = 20

To create a family of knapsack sets with an increasing number of variables, we can add one item at a time with each new variable. So, for n=4, we can add a new item with weight 5 and value 25. For n=5, we can add another item with weight 6 and value 30, and so on.

Now, to ensure that the associated family of minimal covers grows exponentially with n, we need to carefully choose the weights and values of the new items we add. For example, for n=4, we can add an item with weight 8 and value 50, and for n=5, we can add an item with weight 12 and value 100. This way, as n increases, the weight and value of the new items also increase exponentially, leading to an exponential growth in the minimal covers.

In summary, to produce a family of 0,1 knapsack sets with an increasing number of variables, we can gradually add new items with increasing weights and values. This will ensure an exponential growth in the associated family of minimal covers. I hope this helps you in solving the problem. Good luck!