Questions tagged [p-vs-np]

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P vs NP and the Time Hierarchy

Assuming $P\neq NP$, is it possible that there exists a $k$ such that $P\subseteq\textsf{NTIME}(t^k)$? There reason I ask this is that I assume the following: $$P=NP \implies \forall k\ \exists j.\ \...
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248 views

Are there consequences for P ≠ NP that are unintuitive?

It's often regarded that the most intuitive answer to the question of $P$ vs $NP$ is that $P ≠ NP$. This is often illustrated with some consequences that would follow if $P = NP$ were true. Things ...
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54 views

Is anything known about the structure of sets of valuations representable by 3CNF formulas?

Let's suppose we have propositional variables $x_1 ... x_n$. A valuation is an assignment $v$ s.t. $v(x_i)$ is an element of $\{false, true\}$ for $1 \leq i \leq n$. So, there are $2^n$ possible ...
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36 views

Weaker conjectures to prove in order to arrive at P =/= NP

We know we have a long way to go before we come to a proof of P $\neq$ NP. We also know that this road is studded with minor conjectures that will have to be proved/disproved in order to arrive at the ...
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118 views

What can be a Zero Knowledge Proof of a working SAT Algorithm?

Me and my colleague are exploring new ideas to solve SAT efficiently (i.e. in polynomial time) and it's the case that there is a candidate algorithm. Unfortunately, neither of us can write scripts ...
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201 views

Is valid the notion of infinity for the NP-complete problems?

We have defined two complexity classes which have a close relation with the notion of infinity. The first one is: We say that a language $L$ belongs to $UP_{\infty}$ if there exist an infinite ...
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26 views

Are inputs of NP problems too implicit?

I'm reading on wikipedia about P vs NP and I have a question. Consider the zero sum problem: given a set of integer A, determine if there is a non empty subset such that the sum of all its elements is ...
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46 views

Is a or free SAT formula NP complete?

Let $L$ be the languague which contains all satisfiable formulas which do not have the or symbol $\lor $. Or more precise $$L=\{\phi | \phi \text{ is a satisfable formula which is only using the ...
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25 views

$NP$ 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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24 views

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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1answer
346 views

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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1answer
91 views

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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743 views

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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121 views

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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114 views

If NP is easy on average then does it mean P=NP?

If $NP=RP$ then $NP$ is easy on average. Then from point $1$ in abstract in http://lance.fortnow.com/papers/files/derand.pdf which says $NP$ is easy on average implies $P=BPP$ do we have $NP=RP\...
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127 views

How not to prove that P ≠ NP implies NP ≠ PSPACE

Let's define the two variants of the Travelling salesman problem: $TSP_{opt}$ : Give me the shortest tour $TSP_{dec}$ : Is there a tour of $l$ or shorter (Yes/No) Now assume $P \neq NP$: Since $...
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89 views

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 ...
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1answer
34 views

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?
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Time Complexity and Optimization for the Algorithm?

I have found a algorithm to check whether a Hamiltonian Cycle Exists in the graph or not, but not able to compute/analyse it's time complexity. The algorithm is as follows : Label all the vertices ...