Question

Is the following problem NP-Complete?

The Rice bowl problem is to pick the ingredients for your bowl. You are given a set of ingredients I1 to In. Each ingredient Ii comes with
a quality qi and a quantity si . You are also given a bowl size S and a quality goal Q. Can you select a subset of the ingredients that both fit in the bowl (the sum of their
si is at most S) and have enough quality (the sum of their qi is at least Q).

I know that this problem is NP-Complete (the knapsack problem can be reduced to it), but I am having difficulty formulating a proper proof for this.

I know that I have to show that the knapsack problem is in np, which can easily be done with a certificate. Next I have to show that the knapsack problem can be reduced to this problem. Lastly I have to simply show that if knapsack is np complete, then this problem is also np complete.

The problem is I can't seem to formulate a formal proof to do the above. Can someone please help out with that?

Answer 1

The Rice bowl problem has two constraints 1)Quality 2)Weight.

This is similar to a multidimensional knapsack problem. If there is more than one constraint, we get the multidimensional knapsack problem or m-dimensional knapsack problem. The multidimensional knapsack problem is obviously a NP Complete problem.

*In image, m is number of constraints.

Each item j consumes an amount w_ij from each resource i.

For any fixed m>=2, these problems do admit a pseudo-polynomial time algorithm and a PTAS ( polynomial-time approximation scheme ).