So-called easy, or
tractable, problems can be solved by computer algorithms that run in
polynomial time; i.e., for a problem of
size n, the time or number of steps needed to find the
solution is a polynomial function of n. Algorithms for
solving hard, or
intractable, problems, on the other hand, require times that are
exponential functions of the problem size n. Polynomial-time
algorithms are considered to be efficient, while exponential-time
algorithms are considered inefficient, because the execution
times of the latter grow much more rapidly as the problem size
increases.
A problem is called NP (nondeterministic polynomial) if its solution (if one exists) can be guessed and verified in polynomial time; nondeterministic means that no particular rule is followed to make the guess. If a problem is NP and all other NP problems are polynomial-time reducible to it, the problem is NP-complete. Thus, finding an efficient algorithm for any NP-complete problem implies that an efficient algorithm can be found for all such problems, since any problem belonging to this class can be recast into any other member of the class. It is not known whether any polynomial-time algorithms will ever be found for NP-complete problems, and determining whether these problems are tractable or intractable remains one of the most important questions in theoretical computer science. When an NP-complete problem must be solved, one approach is to use a polynomial algorithm to approximate the solution; the answer thus obtained will not necessarily be optimal but will be reasonably close.