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Would it be an polynomial time algorithm to a specific NP-complete problem, or just abstract reasonings that demonstrate solutions to NP-complete problems exist?
It seems that the a specific algoithm is much more helpful. With it, all we’ll have to do to polynomially solve an NP problem is to convert it into the specific NP-complete problem for which the proof has a solution, and we are done.
P = NP: “The 3SAT problem is a classic NP complete problem. In this proof, we demonstrate an algorithm to solve it that has an asymptotic bound of (n^99 log log n). First we …”
P != NP: “Assume there was a polynomial algorithm for the 3SAT problem. This would imply that …. which by ….. implies we can do …. and then … and then … which is impossible. This was all predicated on a polynomial time algorithm for 3SAT. Thus P != NP.”
UPDATE: Perhaps something like this paper (for P != NP).
UPDATE 2: Here’s a video of Michael Sipser sketching out a proof for P != NP