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For satisfiability problem, its time complexity is at least
T(n) = 2 ^ n
where n is the size of the satisfiability function of the problem.
However, satisfiability problem is not a correct 'NP complete' problem.
As for 3Sat or any kSat problem, most of its functions are originally
P satisfiability problem, not 'NP complete'.
For Hamiltonian Problem, let's take a most simple example,
the problem of no Hamiltonian path or circle or they are at the
end of implementation judgement.
The minimun of
n! > 2 ^ n ^ f(n)
vertex pair comparison is required to satisfy 'NP complete'
when n is big enough, where f(n) approach infinity.
In reality, this mimimum is always much larger.
This is the minimum because pairwised connection is known
knowledge you specify, the implementer can not optimize
on it beforehand, and each path can be rejected only after
at least one pair is not connected.
Verification instance can satisfy this, at least theoretically.
Now, the time complexity is in the same order, much greater than
T(n) = n! / 2 ^ n > 2 ^ [n ^ f(n) - n],
where one path time complexity is taken as 2 ^ n instead of n for
the simplicity of expression and f(n) is a polynomial of degree >= 2
when n is reasonably large.
Other 'NP complete' examples can lead to more or less
complex calculation but the order of time complexity is the same,
at least exponential.
A vertex number of 100 will be enough to satisfy above reasoning.
Such a problem has instances no computing algorithm can operate
on it correctly at present.
Some algorithms claimed passing this value are not correct.
Time complexity is based on theory, not on the partial practical
running of the algorithm.
Some exponential time algorithms can run faster than some
polynomial time algorithms only on 'initial range', they fail on
'large sizes', or they do not scale well.
Both of the above classes are not 'NP complete'!
There are many misconceptions in 'P and NP related problems'.
However, for P != NP proof, both of the above samples are
satisfied and valid.
Conclusion:
P != NP is a theorem.
For NP type of problems, in order to design algorithm,
the time complexity only for correctly examing the input can exceed
exponential limit.
Please cite.
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