Questions tagged [np-intermediate]

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Are there any “natural problems” which are known to be NPI under weak assumptions

Are there any "natural problems" which are known to be NPI under weak assumptions. By weak assumptions I mean something like $P \neq NP$ or $NP \neq Co-NP$
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3answers
115 views

The most subtle NP-“intermediate” problem

What is the $NP$ problem whose status in $P$ or $NP$-complete is still unsettled, as of 2018? This question is inspired by the following two recent breakthroughs: The work of Mulzer et. al on $NP$-...
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1answer
59 views

Assuming NP≠coNP, do we have a similar theorem to Ladner's?

We have that $\mathrm{NP\neq coNP\iff NP\neq NP\cap coNP}$. So by assuming that $\mathrm{NP\neq coNP}$, can we prove the existence of intermediate problem between $\mathrm{NP}$-complete and $\mathrm{...
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1answer
1k views

NP-complete problem with a polynomial number of yes-instances?

I have the impression that for every NP-complete problem, for infinitely many input sizes $n$, the number of yes-instances over all possible inputs of size $n$, is (at least) exponential in $n$. Is ...
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0answers
167 views

Complete problems in NP∩coNP

I often read in Complexity literature that NP∩coNP is unlikely to have any complete problems. Is that unlikelihood "proved" ? By proved, I mean that there would be a theorem that would relate the ...
3
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1answer
80 views

Assuming that $P \neq NP$, show that there exist sets $A$ and $B$ in $NP$ such that neither $A \leq _T^p B$ nor $B \leq _T^p A$

My question is as follows: Assuming that $P \neq NP$, show that there exist sets $A$ and $B$ in $NP$ such that neither $A \leq _T^p B$ nor $B \leq _T^p A$, where $A \leq _T^p B$ if there exists a ...
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1answer
123 views

Is following observation on Ladner's theorem correct?

Suppose $NP\subseteq DTIME[n^{f(n)}]$ where $f(n)$ is any function satisfying $\omega(1)$ then is it true $P=NP$ holds? Ladner's theorem states infinite time hierarchy between $P$ and $NP$. That is ...
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1answer
450 views

Are NP-complete sets formed from two other sets only if at least one is NP-hard?

This question is somewhat of a converse to a previous question on sets formed from set operations on NP-complete sets: If the set resulting from the union, intersection, or Cartesian product of two ...
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1answer
116 views

Constructing languages in NPI other than through Ladner's Theorem

I have seen proofs of Ladner's theorem which detail the construction of languages in NPI assuming P $\neq$ NP. However, I was wondering if there are any other constructions using the fact that sparse ...