这是 NP-Hard 问题的特例吗？

Question

I have the feeling that this is an instance of a NP-hard problem but it's been a while since I've been in school I'm having a difficult time verifying which one it would reduce to. If I have a Collection of sales, where each sale has a price (weight) as well as a set products which the sale applies with cardinality > 0 then find a collection of sales with the greatest combined price, where no sale shares a product with any other sale (you cannot have the multiple sales apply to the same product).

Answer 1

Yes, this is a generalization of a known NP-Complete problem: Set Packing .

This is a generalization of the known problem, because of the weights, which could vary in your problem, but if you set weight(sale) = #products_in_sale , you get exactly the Set Packing optimization problem. ¹

This means, your problem is at least as hard as Set Packing, and thus NP-Hard.

(1) This is basically the polynomial reduction, given an instance of set packing, you generate an instance of your problem, with weight(sale) = #products_in_sale , and if the latter is solveable in polynomial time, so is the former.