I just started learning about P vs Np and have a basic understanding of P, NP, and NP complete. I just wanted a little clarification on how each interacts with the others. Since NP-Complete is in NP would that mean if P was found to be equal to NP, problems that are NP-Complete would be polynomial time solvable as well? Or would P also need to equal NP-Hard as well in order for NP Complete to be polynomial time solvable?
2026-03-26 22:13:21.1774563201
If P = NP, would that mean P=NP-Complete?
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Every NP-complete problem is also a NP problem, by definition. Therefore, if NP=P, the same will be true for NP-complete problems.
You can't say the same for NP-hard problems. These problems include all problems harder then NP problems, except for the NP-complete problems. In this case, we have that NP-hard $\cap $ NP = NP-complete. So, even if NP=P, we still will have NP-hard $\neq$ P.
This picture may help you to understand how these concepts interact.