P

The class of algorithms that run in polynomial time is called P.

Algorithms that require more (exponential) time are ``intractable''

Some problems seem to inherently require more time

One class of such problems is Nondeterministically Polynomial (NP), also called polynomial-time verifiable

Obviously, P $\subseteq$ NP, but P $\subset$ NP (or P = NP) is an open question

An NP-Complete problem is in NP and is as hard as any problem in NP.
Such a problem not necessarily in NP is called NP-Hard.

If P = NP, then a large class of NP-Complete problems would have a polynomial-time solution.
Thus, most researchers advocate P $\subset$ NP (P $\neq$ NP)

We would like to know the class to which a problem belongs.