Formal Definition of NP-completeness
A decision problem is NP-complete if:
- is in NP, and
- Every problem in NP is reducible to in polynomial time.
can be shown to be in NP by demonstrating that a candidate solution to can be verified in polynomial time.
Note that a problem satisfying condition 2 is said to be NP-hard, whether or not it satisfies condition 1.
A consequence of this definition is that if we had a polynomial time algorithm (on a UTM, or any other Turing-equivalent abstract machine) for, we could solve all problems in NP in polynomial time.
Read more about this topic: NP-complete
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