TITLE: what is the relationship between the complexity class E(and EXP) and NP?
QUESTION [6 upvotes]: I want to know any relationship between the complexity class E(and EXP) and NP.
I also would like to know whether there is any $DTIME$ formulation or relations of $NTIME(O(n^k))$ where n is the size of input, and k is constant.
edit: for example, is NP contained in EXP?

REPLY [4 votes]: We have:
$$ P \subseteq NP \subseteq PSPACE \subseteq EXP \,, $$
and $P \subsetneq EXP$. The latter is the result of time-hierarchy theorem; see here. Therefore, at least one of the inclusions above is strict, but no one knows which.
Furthermore, we know from Ronald V. Book's 1972 paper that $E \ne NP$.