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Questions about decision problems that can be solved on nondeterministic Turing machines in time polynomial in the length of the input.

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Why do we prove X is in NP to prove it is NP-complete?

The steps to show something is NP-complete is to show it is a member of NP, and then show an NP-complete problem reduces to it in polynomial-time. Why is the first step necessary? … Is it possible to reduce an NP-hard problem to an NP-complete problem in polynomial time? Because that would be the only instance in which this is important. …
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