TITLE: Why are Polynomial Time Problems Considered Tractable, and Larger Times are Not?
QUESTION [9 upvotes]: I've been reading up on $P=NP$, problem tractability, etc. Here's my question:
Why is it that we consider problems that can be solved in polynomial time - or algorithms/problem-solvers running in polynomial time - nice, tractable, solvable, etc. while considering problems with a greater time complexity to be intractable, inefficient to compute, etc.?
My thoughts are that:  

Surely it can't be the case that there is a stark divide
(solvable-unsolvable) between algorithms taking polynomial time and
algorithms taking marginally more than polynomial time.
Couldn't we have an arbitrarily large polynomial expression, such
that this polynomial is slower to compute than some non-polynomial?

REPLY [7 votes]: If you accept that exponential resource use is never tractable, the minimum requirement for bounds defining tractability is that they form a class of increasing functions that are subexponential,  $f(n+1)/f(n) \to 1$. 
If you accept that composition of two tractable algorithms is tractable, the bounds should also be closed under function composition $f(g(n))$ where $g(n)$ is the largest size of output that the first procedure can produce on size $n$ inputs.  To relate this to composition of the resource bounds themselves, the complexity measure should be for time, or include time, so that a bound of $g(n)$ implies at most $g(n)$ bits of output.  Then we simply want the bounds to be closed under function composition.
Time complexity classes satisfying these conditions include polynomial time, and quasi-polynomial time.  Subexponential time, whether in the weak sense $\frac{f(n+1)}{f(n)}\to 1$ or the strong sense $O(2^{n^{o(1)}})$ or the intermediate versions described at Wikipedia, is not composition-closed.
http://en.wikipedia.org/wiki/Quasi-polynomial_time#Quasi-polynomial_time .
Related:
https://cstheory.stackexchange.com/questions/3439/why-do-we-consider-log-space-as-a-model-of-efficient-computation-instead-of-pol/