Is the MinMax/optimization/search variant of a decision problem always easier/equal in complexity because we can always reduce them to their decision variant?
If the longest path problem could be solved in polynomial time, it could be used to solve this decision problem, by finding a longest path and then comparing its length to the number k. Therefore, the longest path problem is NP-hard. It is not NP-complete, because it is not a decision problem.
Is the max-variant of this problem only NP-hard because the decision-variant is also NP-hard (and NP-complete)? (and not because it is harder)
What about polynomial time problems? If the decision variant of a problem is in P, I would assume that the optimization variant is not NP-hard, because that would imply P = NP. Do we say that it is in P then, or is it in another class/no class?