Understanding NP Problems: Proving Yes or No?

[ak416]

Homework Statement

Im just a little confused about the definition of an NP problem. The most common definition I hear is: a decision problem is in NP if it has a polynomial time certification. So if it is a yes or no question, there is a proof of it in polynomial time. But my question is, when proving a decision problem is in NP, do you have to assume the answer is yes and show a polytime certification exists and then assume the answer is no and show a polytime certification of that answer exists? Or do you choose either yes or no and show that a polytime certification of that answers exists? I hope somebody can help me because I have an assignment due tomorrow asking me whether certain problems are in NP.

Homework Equations

The Attempt at a Solution

 

[Valkarie]

To prove a decision problem is in NP, you will have to show that both answers (yes and no) can be certified in polynomial time. This means that you need to provide a verification procedure that takes polynomial time to verify a given solution. For example, if the decision problem is whether a number is prime or not, you would need to provide a verification procedure that takes polynomial time to check if the given number is in fact prime.

 

[piercas]

it is important to have a clear understanding of the definitions and concepts related to your field. In this case, understanding NP problems and their proofs is crucial for anyone working in the field of computer science or mathematics.

To answer your question, when proving a decision problem is in NP, you do not have to assume the answer is yes or no. You instead have to show that there exists a polynomial time algorithm that can verify a given solution to the problem. This algorithm should be able to check the validity of the solution in polynomial time, regardless of whether the answer is yes or no.

In other words, to prove a decision problem is in NP, you have to show that there exists a polynomial time certification for both a "yes" and a "no" answer. This means that the problem can be solved in polynomial time, whether the answer is yes or no.

I understand that this may seem confusing, but it is important to remember that NP problems are decision problems, meaning they have a yes or no answer. The polynomial time certification is simply a way to verify the solution to the problem in a reasonable amount of time.

In conclusion, when proving a decision problem is in NP, you do not have to choose either yes or no. You have to show that there exists a polynomial time algorithm that can verify a given solution, regardless of the answer being yes or no. I hope this helps clarify the concept of NP problems for you. Good luck with your assignment!