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Concept
NP (complexity)
Summary
In computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems. NP is the set of decision problems for which the problem instances, where the answer is "yes", have proofs verifiable in polynomial time by a deterministic Turing machine, or alternatively the set of problems that can be solved in polynomial time by a nondeterministic Turing machine.
NP is the set of decision problems solvable in polynomial time by a nondeterministic Turing machine.
NP is the set of decision problems verifiable in polynomial time by a deterministic Turing machine.
The first definition is the basis for the abbreviation NP; "nondeterministic, polynomial time". These two definitions are equivalent because the algorithm based on the Turing machine consists of two phases, the first of which consists of a guess about the solution, which is generated in a nondeterministic way, while the second phase consists of a deterministic algorithm that verifies whether the guess is a solution to the problem.
It is easy to see that the complexity class P (all problems solvable, deterministically, in polynomial time) is contained in NP (problems where solutions can be verified in polynomial time), because if a problem is solvable in polynomial time, then a solution is also verifiable in polynomial time by simply solving the problem. But NP contains many more problems, the hardest of which are called NP-complete problems. An algorithm solving such a problem in polynomial time is also able to solve any other NP problem in polynomial time. The most important P versus NP (“P = NP?”) problem, asks whether polynomial-time algorithms exist for solving NP-complete, and by corollary, all NP problems. It is widely believed that this is not the case.
The complexity class NP is related to the complexity class co-NP, for which the answer "no" can be verified in polynomial time. Whether or not is another outstanding question in complexity theory.
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