# Questions tagged [p-vs-np]

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### What is the simplest known NP-Complete problem for testing P=NP solutions? [closed]

About a year and a half ago I asked this question regarding $\mathsf{P}=\mathsf{NP}$. The answers have helped me understand the problem tremendously and since then I've dabbled further into the topic....
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### P Vs NP can never be proven one way or the other!

Okay, so I was reading a bit about P vs NP problems. And I found out that proving P Vs NP is an NP problem. And since if we prove that any problem in NP is a P that would mean that we have NP=P. ...
2k views

### If a problem is in P solved via dynamic programming, is it also in NP?

So I can solve a given problem using dynamic programming in $O(n^2k^2)$ time complexity. This means that the problem is in P. But I am asked if it is in NP. My answer is, "Since it is also polynomial ...
31 views

### Does computational complexity theory take into account problems with subjectivity in the verification of a solution?

When we discuss P vs NP we are looking at the difference between problems that are easily solved versus easily verified (wrt polynomial vs exponential time). But in both cases these are black-and-...
213 views

### What's the flaw in the P != NP proof in the article “The Computational Complexity of the Traveling Salesman Problem”

I am reading through some proof of inequality of P and NP but they are not accompanied by the flaws in the reasoning so I'm trying to find them by myself, just to see if I'm getting the logic right. ...
169 views

### P vs. NP and Godel's Theorems

This post is based loosely on a previous post, but the presentation is somewhat different and hopefully much more succinct. Basically, I'm wondering if it is plausible for there to be a formal proof ...
88 views

### How to prove Exact cover problem is NP Complete using Vertex Cover problem?

Using reduction theorem in NP, we want to prove that Exact cover is NPC by reducing it from Vertex Cover Problem. It is easy to derive it from SAT, but we can't find a solution yet to derive it from ...
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### Is this statement of P = NP in Agda correct?

Looking for a self-contained statement of P = NP in type theory, I stumbled upon this short Agda formalization (a cleaned up version is reproduced below). The statement here does seem to express the ...
34 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 ...
204 views

### Is it possible that P vs NP is not the real problem?

Lets assume that I found a polynomial solution for Hamiltonian path problem. It is known that you can reduce this problem to SAT. How ever it will be a special case of SAT. Just the case where there ...
5k views

### Evolving artificial neural networks for solving NP problems

I've recently read a really interesting blog entry from Google Research Blog talking about neural network. Basically they use this neural networks for solving various problems like image recognition. ...
43 views

### How is $\text{PCP}[O(\log n),O(1)]$ NOT P?

As a prover, we just try to convince the verifier that it's correct, no matter whether it is or not. So we can just analyze every possible route. For $\text{PCP}[O(\log n),O(1)]$, won't there just be ...
116k views

### What is the definition of P, NP, NP-complete and NP-hard?

I'm in a course about computing and complexity, and am unable to understand what these terms mean. All I know is that NP is a subset of NP-complete, which is a subset of NP-hard, but I have no idea ...
3k views

### Would proving P≠NP be harder than proving P=NP?

Consider two possibilities for the P vs. NP problem: P=NP and P$\neq$NP. Let Q be one of known NP-hard problems. To prove P=NP, we need to design a single polynomial time algorithm A for Q and prove ...
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### Confusion about P versus NP [duplicate]

I'm sure that in my following question my reasoning is extremely simplistic and flawed, but I think if someone answered this it would help me understand what the P vs NP conundrum is. So here is my ...
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### P/NP - Polynomial Reduction vs Certificate

I am learning about the P/NP problem right now, and I don't understand when to use polynomial reduction and when to use a certificate. How I understand polynomial reduction is that you can use it to ...
151 views

### Baker, Gill, Solovay - construction of oracle B such that P^B != NP^B

I have some questions about Baker, Gill, Solovay proof of the existence of an oracle such that P^B != NP^B. The proof can be found in Siam Journal of Computing, 4:432-442, 1975 . Why Isn't this ...
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### How not to solve P=NP?

There are lots of attempts at proving either $\mathsf{P} = \mathsf{NP}$ or $\mathsf{P} \neq \mathsf{NP}$, and naturally many people think about the question, having ideas for proving either direction....
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How am I supposed to read the P=?NP relativization proof? I am reading the classical paper Relativization of the P=?NP problem by Baker, Gill and Solovay, in particular the proof that there exist an ...
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### Is the P vs NP problem, an NP or co-NP problem?

A few years ago, a youtube channel named hackerdashery, made an extraordinary youtube video explaining P vs NP, in a semi-vulgarized way : https://www.youtube.com/watch?v=YX40hbAHx3s At 7 minutes ...
444 views

### One-way Trapdoor Function

Do the functions of the collatz conjecture and their inverses model a Trapdoor Function? If given a, b, a^-1, b^-1 and your choice of f(x), is it “hard in the average case” to find some secret x? I ...
138 views

### P vs NP question from GeeksforGeeks

From here: https://www.geeksforgeeks.org/algorithms-np-complete-question-2/ Let S be an NP-complete problem and Q and R be two other problems not known to be in NP. Q is polynomial time reducible to ...
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### Contradiction proof for inequality of P and NP?

I'm trying to argue that N is not equal NP using hierarchy theorems. This is my argument, but when I showed it to our teacher and after deduction, he said that this is problematic where I can't find a ...
102 views

### proving that $P\ne NP$ under an assumption

Suppose that $P^{SAT} \not\subseteq coNP$. Prove that $P\ne NP$. What I did: Suppose that $P=NP$. Then, $P = coP = NP = coNP$. We know that $P^P = P$. Then, by assumption: $P^{NP} = NP = coNP$ ...
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### A tricky P=NP problem

Define an operator $\pi(\cdot)$: for a language $L$, $\pi (L)$ is the set of all prefixes of strings in $L$ with length at least half of the original string. Prove that if $\mathsf{P}$ is closed under ...
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### Reducing SAT to a P problem in polinomial time [duplicate]

Does reducing SAT in polynomial time to a P problem would mean that P = NP?
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### If $\mathbf{P} = \mathbf{NP}$, then is $\mathbf{L} = \mathbf{NL}$?

If $\mathbf{P} = \mathbf{NP}$, then is $\mathbf{L} = \mathbf{NL}$? I am asking this question because, for other non-deterministic classes, it seems $\mathbf{P} = \mathbf{NP}$ always establishes that ...
83 views

### What is Ironic complexity? What are some good resources to learn about it?

The term "Ironic complexity" was coined by Scott Aaronson for the stuff Ryan Williams does in the area of complexity theory. Could anyone tell me what kind of problems and approaches does Ryan ...
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### If graph isomorphism is in P, is then P = NP?

I think that, since graph isomorphism is not known to be $\textbf{NP}$-complete, we can not reduce all problems in $\textbf{NP}$ to it, and therefore the implication does not hold. Additionally, in ...
2k views

### What are the implications of P=NP? [duplicate]

Is there a list of implications of $P=NP$? Presumably, a proof of $P \ne NP$ will be by contradiction, for which a list of consequences of $P=NP$ would be useful.
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### Effective Procedures and P vs NP Problem

If, suppose, P doesn't equal NP. Implication of this statement is that there is no effective procedure to solve a hard problem; however there exists an acceptable solution S. I have following two ...
224 views

### Why any problem can be reduced to SAT is NP-Complete?

I have a book statement says the title, I don't understand it. From my current understanding if a problem A can be reduced to a problem B then it only means B is at least as difficult as A.
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### Given that A reduces to B in $O(n^2)$ and B is solvable in $O(n^3)$, solve A

Suppose a problem A reduce to problem B and reduction is done in $O(n^2)$ time. If problem B is solved in $O(n^3)$ time then what about the time complexity of problem A? Approach: A is reduced ...
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### Can we detect perfect matchings in P? in NP? in coNP?

This question concerns the classes P and N P . If you are familiar with them, you may skip the definitions and go directly to the question. Let L be a set. We say that L is in P if there is some ...
278 views

### 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.\ \...
75 views

### Are there problems that are DSPACE(O(1)) complete?

I know there are problems that are NL-complete, NP-Complete, PSPACE-complete, etc. Are there problems that are DSPACE(O(1))-complete I.e. NSPACE(O(1))-Complete I.e. Reg-Complete? Thanks!
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### 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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### Does indirect diagonalization a relativize technique?

My main question is can with R.kanon , Fortnow ,... technique that shows lower bounds for SAT seperate P and NP ? Baker-Gill-Solovay showed that $P?=NP$ could not be solved with relativization. Does ...
429 views

### Chomsky Hierarchy and P vs NP

I have read multiple questions here that involve this kind of subject but I haven't found any definite answer. In what class do regular languages belong? (P or NP or some regular are P and other NP), ...
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### Are there infinite possibilities to the outcome of the P vs. NP question?

The P vs. NP poll provides 3 possibilities: equal, not equal, and independent. This is reasonable, because despite the law of the excluded middle you must supply a proof for your answer, which itself ...
210 views

### Logical characterization of P versus NP problem (and references for least fixed point logic)

Wikipedia says the following (and more) about the logical characterization of the P versus NP problem here: Thus, the question "is P a proper subset of NP" can be reformulated as "is existential ...
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### What's wrong with the following argument that $NP \subset coNP$? [duplicate]

What's wrong with the following argument that $NP \subset coNP$? let $L \in NP$; then there exists an NTM $N$ that decides $L$ in $f(n)$ time where $f(n) = O(n^k)$ for some natural number $k$. ...
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### Does Provable P equal Provable NP?

My question is a very basic one. It seems feasible to believe that $\mathsf{P = NP}$, because there is some "pathological" good algorithm for SAT, yet it is impossible to prove that the algorithm is ...
321 views

### P vs NP problem (Student example)

Hello dear stackexchangers, I have a simple question, and I would like to say that I am not a scientist. When I read the problem statement on this link: http://www.claymath.org/millennium-problems/p-...
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### P vs NP and can an oracle make P=EXPTIME?

As I understand, diagonalization cannot be used to prove or disprove P vs NP, because for some oracle $A$, $P^A = NP^A$, whereas under another oracle $B$, $P^B \neq NP^B$. I don't fully understand ...
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### Independence Implies P $\ne$ NP

Suppose P vs. NP is independent of ZFC. Then there cannot be an efficient SAT solver, otherwise it would constitute a proof for P = NP. Therefore P $\ne$ NP. What we see here is that independence ...
765 views

### Why doesn't Godel's Second Incompleteness Theorem rule out a formalizable proof of P!=NP?

I'm sure there must be something wrong with the following reasoning because otherwise a lot of P vs. NP research would be curtailed but I cannot determine my error: For any fixed integer $k>0$ ...
158 views

### How are boolean circuits used for solving P vs NP?

In the paper https://web.stanford.edu/~gavish/documents/sipser-pvsnp.pdf , it is mentioned under the Status section that boolean circuits have been used to try and solve P vs NP. Can anyone explain to ...