Maybe this is very obvious, but if we had an algorithm in P (so this algorithm gives a yes/no answer in polynomial time), is there a more efficient way to find the solution beyond just guessing and checking?
So, suppose SAT is in P (I know this is an NP-Complete problem, but this seems like the best example for what I'm trying to ask). This means that there is a polynomial time algorithm that will tell you yes or no depending on whether or not the given input is satisfiable.
It would seem that there should thus be an efficient way to find/extract this satisfying assignment (rather than just know it exists, if there is one). However, I can't think of any efficient way to utilize this poly-time algorithm to find such an assignment.
** side note ** For maximization/minimization (e.g. Knapsack) problems I know that you can use binary search to find your solution, but my question is more pertaining to these non-maximization type problems like SAT
You don't have to guess the entire thing and then test it.
You can get a satisfying valuation (if it exists) like this:
Pick a variable, make it false, remove it from all clauses/remove satisfied clauses. Consult the SAT oracle, which apparently runs in polynomial time today. If it's still satisfiable, fine, keep it. Otherwise it must be true, restore the clauses and clean up the clauses again. There's no backtracking, there's just one call to SAT for every variable. That whole thing is still in polynomial time.
Or if that's what you had in mind, then well, that's it. Does it really matter though? Polynomial time is polynomial time, and this isn't usable in practice anyway so wall-clock time is hardly a concern.