NP (complexity)

NP (complexity)

In computational complexity theory, NP is one of the most fundamental complexity classes. The abbreviation NP refers to "nondeterministic polynomial time."

Intuitively, NP is the set of all decision problems for which the instances where the answer is "yes" have efficiently verifiable proofs of the fact that the answer is indeed "yes". More precisely, these proofs have to be verifiable in polynomial time by a deterministic Turing machine. In an equivalent formal definition, NP is the set of decision problems where the "yes"-instances can be decided in polynomial time by a non-deterministic Turing machine. The equivalence of the two definitions follows from the fact that an algorithm on such a non-deterministic machine consists of two phases, the first of which consists of a guess about the solution which is generated in a non-deterministic way, while the second consists of a deterministic algorithm which verifies or rejects the guess as a valid solution to the problem.

The complexity class P is contained in NP, but NP contains many important problems, the hardest of which are called NP-complete problems, for which no polynomial-time algorithms are known for solving them (although they can be verified in polynomial time). The most important open question in complexity theory, the P = NP problem, asks whether polynomial time algorithms actually exist for NP-complete, and by corollary, all NP problems. It is widely believed that this is not the case.

Read more about NP (complexity):  Formal Definition, Introduction, Why Some NP Problems Are Hard To Solve, Equivalence of Definitions, Relationship To Other Classes, Other Characterizations, Example