I want to reduce this variation of knapsack problem to my own problem.

Hi Max, I have a follow up question. If I have one more constraint on $B$ which is $\sum_{h=1}^i (w_h+v_h)/2 =B$, is this problem still NP-hard?

So is my problem just a special case of the generic knapsack problem you formulated with c=1? If so, it may not be sufficient to prove the NP-hardness of my knapsack problem. Is it correct?

Another possible issue is that the number will goes very big such that 64 bits machine cannot represent when the size of S is large.

Thanks for your answer. The function is quite simple. The only issue is the length of bits increases linearly with the size of S'. Can we find a good way to reduce the memory cost (e.g., increasing sub-linearly)?

@MartinSleziak I post this question after I have look at the question you link to in your comments. The solution for that question is still very costly. I allow values in S to satisfy some requirements or relationship. I hope that it could make the problem easier.

@MartinSleziak Thanks for your suggestions. I have deleted my old question posted in MATH SE.

@MartinSleziak Thanks. I was about to say this. I know there is a similar question link posted many years ago. But my case is different, I allow values in S to satisfy some requirements or relationship. I hope that it could make the problem easier.