Assuming there is no polynomial algorithm for a general NP problem. There might be an alternative.
For example. Consider the Travelling salesman problem on N nodes.
There may be a sequence of algorithms:
$$A_1,A_2,A_3,A_4,....$$
Such that the running time $T(A_n) \in \mathbf{P}$
The way this would be possible if the general algorithm $G_n$ which generates the algorithm $A_n$ is such that $T(G_n) \in \mathbf{NP}$.
Thus the general algorithm can generate a sequence of algorithms that together form a polynomial time sequence yet the general algorithm is in $\mathbf{NP}$ because it takes non-polynomial time to generate the algorithms.
So for example, the general algorithm $G_{100}$ might take a very long time to generate an algorithm $A_{100}$ but once we have the algorithm $A_{100}$ it might be able to solve the travelling salesman problem on 100 nodes very "quickly".
Equally, it may be possible for the $G'_{100}$ to generate an algorithm $A'_{100}$ to efficiently factor 100 digit numbers. (The algorithm might be huge and take up a lot of memory)
Therefor I'm wondering if anyone has searched for or catalogued for example a sequence:
$$A_1, A_2, A_3,...$$
of algorithms for specific cases of things like travelling salesman, or boolean satisifiability problems, since it seems like there would be value in having algorithms for each specific value of $N$ (length of input). Which might have greater practical use than a general algorithm.
An interesting conjecture might be something like "There exists a sequence of polynomial time algorithms for NP problems, yet the sequence of algorithms can only be generating in non-polynomial time." Do you know if such a conjecture exists?