# Questions tagged [p-vs-np]

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### $NP$ is not in $P(n^k)$ for any fixed $k \geq 1$

I encountered this problem which asks to show that for any fixed $k \geq 1$, $NP$ is not contained in $P(n^k)$... As an attempt, I thought of using the time hierarchy theorem which says that there ...
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### How sub-exponential time does $\text{3SAT}$ have to be to make $\text{NP} \neq\text{EXP}$? What else would imply $\text{NP} \neq\text{EXP}$?

The exponential-time hypothesis posits that if $\mathsf{3SAT}$ has NO subexponential time algorithm (i.e. one in $\mathcal O(2^{o(n)})$), then $\mathsf{P}\neq \mathsf{NP}$. However, I am interested ...
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### Prove or disprove, If A ≤p B and B is NP-hard, then A is in NP-hard

Intuitively if A can reduce to B, and B is NP-Hard, A might be NP Hard but maybe not. If there is a way to solve A that does not involve reducing to B, it might be faster. How do I formally disprove ...
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### Prove or Disprove, 3SAT ≤p 2SAT, then P = NP

I know that 3SAT is in NP and 2SAT is in P. And 2SAT can reduce to 3SAT just says 3SAT is strictly harder than 2SAT, so I don't think this proves P = NP, but it doesn't seem to disprove it either.
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### Show that P is a subset of NP

Okay, before I start with the question I would like to point out that I am aware that there are many proofs of this question online. However, I am interested in showing this with the definitions of my ...
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### Consequences from a lower bound of SAT problem

I'm not sure how lower bounds affect the question to the P=NP problem. I.e. : Let a SAT instance with a size of n be transformed into an instance of a problem X with a size of n3. If you find a ...
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### Using the following algorithm, is it possible for P to equal NP?

So this question is a follow up from this post: Quantum Circuits as a Path Finding Algorithm where I contemplated the possibility that a well designed quantum circuit could be be described as a BST ...
### If X is in NP then $\overline{X}$ is in NP. True, false or “we don't know”? Why?
If X is in NP then $\overline{X}$ is in NP. True, false or "we don't know"? Why?