各种物品的背包

Question

Suppose a regular knapsack problem: you have a weight constraint, C , the number of items with Value and Weight (V, W) . You want to maximize the V while W being under the C . In this question, you can only have one of each item.

But there is an additional twist to the problem. You want to have a variety of items. Suppose the question states that you want to have at least 5 (or any number) of different items. If a solution has any less than 5 different items, the answer is not valid. Is there an approach to this problem that solves this?

Answer 1

It's just another form of constraint, so lets see how Weight (at most C weight) and Diversity (at least 5 different items) differ:

• Weight starts as a valid (empty bag is below C), while Diversity starts as invalid (empty bag doesn't contain 5 different items).

The first thing to note is, that with the additional constraint you need a notion of invalid / unsolvable, because if there aren't 5 different items that satisfy the weight constraint, there is no solution.

Once you defined a way to return some invalid result, it's really close to the standard knapsack problem. In recursion, just pass remaining allowed Weight and current Diversity. In the recursion anchor case, check Diversity and return invalid if Diversity does not meet requirements, otherwise return the result like in the normal knapsack problem. Check recursion results for invalid and treat it accordingly.