I want to ask a question concerning some aspects of the P vs. NP problem.
There are some results that people use to support a preference / belief about the conjecture $P \neq NP$ (e.g. separation of P and NP over other computation models or fields, some relativized versions, the inability of many people to find polynomial algorithms for any NP-complete problem, etc.)
My question is (not sure how to phrase it exactly):
What if for every "instance" of an NP-complete problem (e.g. 3-SAT) there exists a polynomial algorithm but there is not one fixed polynomial algorithm that solves all instances?
For example there are polynomial-time algorithms for special cases of a (generally) np-complete problem, etc..
Are these not indications about a possible P=NP result?
Are there results that point towards this direction?
What will be the implications for P vs. NP if that hypothesis (or does not hold)? Would it mean that the current context of P vs. NP may need revision and re-framing to capture these alternatives?
Possible sketch of proof that P=NP
If the PH collapses, it collapses as a whole (due to "symmetry of construction" of each level). It seems awkward that at some level the hierarchy collapses while not at other levels since by construction no such foundamental difference exists.
Show that that at some level there can be collapse.