P != NP question

Question

Not a \'pure\' programming question, but since it is deeply involved in programming theory, I thought it best to ask here.

Regarding the P NP problem, this excerpt from

Answer 1

Whether they're related or not is one of the Claypool Foundation's "Millenium Problems", and they'll give a million dollars to somebody providing an appropriate proof that hold up under a few years of intense examination.

The types of questions are more related than they look, since another definition of an NP problem is one that can be solved efficiently with an arbitrarily parallel computer.

One thing that really interests people is the lack of a proof. There are proofs for similar-looking questions, but not this one. That intrigues people, especially mathematicians, since a proof is likely to bring a lot of insight into other things. That is apparently the case for Perelman's proof of the Poincare Conjecture, another of the Millenium Problems.

Another issue is the impact this could have. Right now, few people believe that there is an efficient method for solving NP-complete problems, so a discovery that P!=NP would have little practical impact. A discovery of an efficient way to solve NP-complete problems would revolutionalize a lot of computer science. It would make a lot of things much easier, and would destroy cryptography as we know it by making decryption easy.

Answer 2

Let's assume I am handed a solution to a "hard" problem by a magician, and I can easily verify if this solution is correct or not. BUT, can I compute this solution myself easily? (polynomial time)

This is exactly the question.