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4
votes
2answers
583 views

Why does Schaefer's theorem not prove that P=NP?

This is probably a stupid question, but I just don't understand. In another question they came up with Schaefer's dichotomy theorem. To me it looks like it proves that every CSP problem is either in P ...
2
votes
1answer
93 views

$P \neq NP$ and determinism

Suppose $P \neq NP$. Does it imply that there exists some superpolynomial time bound, such that any $NP$-complete problem, like SAT, can be used to simulate an arbitrary deterministc Turing Machine ...
5
votes
1answer
108 views

Does $P \neq NP$ imply $NP \neq PSPACE$?

Is it true that $\mathsf{P} \neq \mathsf{NP}$ implies $\mathsf{NP} \neq \mathsf{PSPACE}$? I have here some problem that is in $\mathsf{PSPACE} \setminus \mathsf{NP}$ if $\mathsf{P} \neq \mathsf{NP}$. ...
1
vote
0answers
64 views

A paper argumenting that P might be equal to NP [closed]

It seems like most serious computer scientists believe that P is not equal to NP, but they just do not know how to prove it. Is there any worth-mentioning paper in which an argument is made in favor ...
1
vote
1answer
51 views

If P is equal to NP, then what happens to the problems those can be solved in polynomial time?

Suppose that an algorithm $A$ is able to solve a problem in NP in polynomial time. Does this effect the good old sorting, searching, shortest path, minimum spanning tree etc. algorithms? Can this ...
2
votes
3answers
79 views

Could an NP-Hard problem be in P in after a basis transform? [closed]

I'm aware that there must be something wrong with my reasoning, but I'm not sure what and neither are a few other CS people I've asked. So here goes: Take the following problem for example: Let ...
11
votes
1answer
147 views

Why is this argument for $P\neq NP$ wrong?

I know its silly, but i managed to confuse myself and i need help settling this Suppose $P=NP$, then clearly for every oracle $A$ we have $P^A=NP^A$ which contradicts the fact that there exists some ...
33
votes
8answers
4k views

What would be the real-world implications of a constructive $P=NP$ proof?

I have a high-level understanding of the $P=NP$ problem and I understand that if it were absolutely "proven" to be true with a provided solution, it would open the door for solving numerous problems ...
2
votes
1answer
129 views

If P = NP, why does P = NP = NP-Complete? [duplicate]

If P = NP, why does P = NP also then equal NP-Complete? I.e. Why would it then be the case that ...
5
votes
6answers
1k views

How is it valid to use oracles in mathematical arguments?

Oracles do not exist. If one did exist, then you would replace them with a subroutine with computational requirements and you would no longer need an "Oracle". Thus, Oracles do not exist almost by ...
1
vote
1answer
73 views

Can oracle arguments separate P and NP?

I know that the general consensus among CS researchers is that non-relativizing techniques will be needed to separate P and NP. However, if there is an oracle language $A \in \textbf{P}$ such that ...
2
votes
1answer
37 views

What is the implication of the sentence: “if any NP complete problem is p time solvable, then all problems in NP are p time solvable”

I find this quote here on page 13 Does it mean that out of all different problems that are NP complete, if any problem is found to have a p time solution, then all the NP complete problems are p ...
1
vote
1answer
240 views

Subset sum algorithm in O(n³ log n)?

I think that I have found an algorithm which resolve exactly the subset sum problem in $O(N^3)$ in the worst case, only for positive numbers. After my research, I'm lost between all the algorithms ...
6
votes
1answer
149 views

If one shows that UNIQUE k-SAT is in P, does it imply P=NP?

Valiant & Vazirani proved SAT is reducible to UNIQUE SAT under randomized probabilistic reductions in polynomial time. Calabro et al. showed that UNIQUE k-SAT is as hard as k-SAT. Now the ...
0
votes
0answers
63 views

Homomorphism erasing information

I would be grateful if anyone could help me with the tricky exerciese *7.52 from Sipser's Introduction to the Theory of Computation 3rd ed. I got stuck in proving that, if P is closed under ...
0
votes
1answer
84 views

Problem with my proof that NP = coNP?

Is there a problem with this proof that NP = coNP? It suffices to show that Satisfiability can be solved efficiently with at most a polynomial number of queries to an oracle for Tautology. The ...
1
vote
1answer
77 views

What are the current known implications of the complexity of Integer Factorization?

According to my limited knowledge we know that since Integer Factorization lies in the intersection of NP and co-NP it cannot be NP-complete unless NP=co-NP. However, do we know any other ...
1
vote
1answer
52 views

A detail on variant of Mahaney's theorem about reductions of sparse languages vs P/NP

Wikipedia states on sparse languages that There is a Turing reduction (as opposed to the Karp reduction from Mahaney's theorem) from a NP-complete language to a sparse language iff NP $\subseteq$ ...
2
votes
2answers
187 views

Does this mean $P = NP$

I am not a formally trained guy on Complexity theory, but due to interest I am learning it. Based on different feedbacks, I have started my journey with Micheal Sipser's "Theory of Computation" (2013 ...
1
vote
2answers
160 views

If P != NP, then 3-SAT is not in P

I hope I'm in the right section: I know that if P = NP, then 3-SAT can be solved in P (Cook), but is the opposite valid, too? If P != NP, then 3-SAT is not in P? Thanks!
0
votes
0answers
26 views

Can P vs NP be independent of accepted axioms? [duplicate]

On wikipedia's page on P vs NP it says that think that of 151 researchers surveyed, their thoughts were as follows: "126 (83%) believed the answer to be no, 12 (9%) believed the answer is yes, 5 ...
5
votes
0answers
100 views

research on OR and AND compression in SAT formulas [closed]

this is a new/advanced paper on OR and AND compression of SAT formulas, a newer area of research that seems not covered in any textbooks so far. A simple proof that AND-compression of NP-complete ...
-4
votes
1answer
600 views

If I solve hard instance, therefore I prove NP=P? [duplicate]

If someone (off-topic) asks a question (on-topic) like this: Suppose that he claims that $\mathcal{P=NP}$. Suppose that someone else (on-topic) gives him an instance of an NP-complete problem that ...
-1
votes
1answer
100 views

what are the basic/typical/common mistakes in P=NP claims? [duplicate]

the P vs NP problem attracts a lot of attention, not all of it desirable, for a wide variety of reasons. there are many P=NP claims eg on this widely cited list maintained by mathematician Woegeorgi, ...
0
votes
0answers
80 views

Assume that $\mathsf{NP} \subseteq \mathsf{P}/\text{log(n)}$, does it imply that $\mathsf{P} = \mathsf{NP}$? [closed]

I am trying to either prove or refute the claim mentioned in the title. Any ideas ?
10
votes
1answer
226 views

Runtime bounds on algorithms of NP complete problems assuming P≠NP

Assume $P\neq NP$. What can we say about the runtime bounds of all NP-complete problems? i.e. what are the tightest functions $L,U:\mathbb{N}\to\mathbb{N}$ for which we can guarantee that an optimal ...
4
votes
2answers
162 views

Is this language depending on P = NP recursive?

Nobody yet knows if ${\sf P}={\sf NP}$. Let us consider the following language $$L = \begin{cases} (0+1)^* & \text{ if ${\sf P}$ = ${\sf NP}$} \\ \emptyset &\text{ otherwise}. \end{cases}$$ ...
0
votes
1answer
251 views

What makes it so difficult to prove P =/≠ NP? — The subset sum issue [closed]

I can't understand or imagine some fact about NP-hard problems. If I understand it correctly there is only one polynomial-time algorithm needed – for whichever NP-complete problem – to ...
3
votes
1answer
142 views

Provability of NP /= P?

I'm a novice to the topic of provability so bear with me... During a discussion with a friend, the question came up whether it could be possible that proving that $NP \neq P$ (or $NP = P$) is an ...
4
votes
1answer
127 views

Existence of NP problems with complexity intermediate between P and NP-hard

Assuming P!=NP, there is a result that there are decision problems intermediate between P and NP-complete. That is, the class NP cannot be a union of two disjoint subsets: P and NP-complete. I could ...
1
vote
1answer
35 views

Complexity class of Determining Hamiltonian cycle

I Know that determining Hamiltonian cycle in a graph is NP complete. For the sake of my clarification, I just want to know that whether the problem remains NP complete with following restrictions ? ...
-5
votes
1answer
103 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 ...
-1
votes
1answer
82 views

Why is SAT not in P? [duplicate]

I'm studing P and NP complexity classes. I like know, why is SAT not in P? Is it because I can not determine if any Boolean expression is satisfiable?
1
vote
1answer
2k views

Proving that if coNP $\neq$ NP then P $\neq$ NP

I am new in complexity theory and this question is a part of a homework that I have and I am stuck on it. Let ${\sf coNP}$ be the class of languages $\{\overline{L}: L \in {\sf NP} \}$. Show ...
2
votes
1answer
1k views

Reduction from Vertex Cover to an Independent Set problem

Assume there exists some algorithm that solves vertex cover problem in time polynomial in terms of $n$ and exponential for $k$ with the run time that looks like this $O(k^2 55^k n^3)$. Can we claim ...
1
vote
1answer
150 views

How to prove polynomial time equivalence?

Define the problem $W$: Input: A multi-set of numbers $S$, and a number $t$. Question: What is the smallest subset $s \subseteq S$ so that $\sum_{k \in s} k = t$, if there is one? (If not, ...
7
votes
4answers
3k views

Flaw in my NP = CoNP Proof?

I have this very simple "proof" for NP = CoNP and I think I did something wrongly somewhere, but I cannot find what is wrong. Can someone help me out? Let A be some problem in NP, and let M be the ...
6
votes
1answer
197 views

$1+\epsilon$ approximation for inapproximable problems

I am currently confused by the following situation: 1) The metric $k$-center problem is inapproximable in polynomial time within $2-\epsilon$ unless $P=NP$. 2) The metric $k$-center problem can ...
3
votes
2answers
760 views

Is the open question NP=co-NP the same as P=NP?

I'm wondering this based on several places online that call $\sf NP=$ co-$\sf NP$ a major open problem... but I can't find any indication as to whether or not this is the same as $\sf P=NP$ problem... ...
3
votes
1answer
91 views

Is it necessary for NP problems to be decision problems?

Professor Tim Roughgarden from Stanford University while teaching a MOOC said that solutions to problems in the class NP must be polynomial in length. But the wikipedia article says that NP problems ...
88
votes
5answers
31k views

In basic terms, 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 ...
0
votes
1answer
171 views

Recusively Enumerable or Recursive dependent on whether P=NP

If a language is defined such that $L = (0+1)^{\ast}$ if $\mathsf{P} = \mathsf{NP}$ and $\emptyset$ otherwise Then $L$ is a regular language if $\mathsf{P} = \mathsf{NP}$, otherwise it is the ...
3
votes
1answer
249 views

If NP $\neq$ Co-NP then is P $\neq$ NP

Does the proof of the widely believed result P $\neq$ NP depend on the proof of NP $\neq$ Co-NP ?
8
votes
3answers
2k views

Proving P = NP without mathematical statements / computer program

This is my first post after being a passive user for some time now. I wish to ask some questions if I may. I am not a mathematician but my question relates to the field of maths/computer science. In ...
8
votes
3answers
1k views

Does P != NP imply that |NP| > |P|?

Is it possible that $\mathsf{P} \not = \mathsf{NP}$ and the cardinality of $\mathsf{P}$ is the same as the cardinality of $\mathsf{NP}$? Or does $\mathsf{P} \not = \mathsf{NP}$ mean that $\mathsf{P}$ ...
2
votes
2answers
165 views

Implications of polynomial time reductions

I'm reviewing for finals and have a sample problem that I think I understand, but would like someone to bless my understanding or smack me and tell me why I'm wrong. I'm presented with a problem ...
5
votes
1answer
269 views

What would an exponential reduction from an NP-complete problem to P signify?

Taking an NP-complete problem like vertex cover if we can find a reduction which is exponential and not polynomial and the reduction we do to a problem can be solved in polynomial time, then what ...
9
votes
3answers
277 views

Proving that if $\mathrm{NTime}(n^{100}) \subseteq \mathrm{DTime}(n^{1000})$ then $\mathrm{P}=\mathrm{NP}$

I'd really like your help with proving the following. If $\mathrm{NTime}(n^{100}) \subseteq \mathrm{DTime}(n^{1000})$ then $\mathrm{P}=\mathrm{NP}$. Here, $\mathrm{NTime}(n^{100})$ is the class of ...
11
votes
2answers
708 views

How can P =? NP enhance integer factorization

If ${\sf P}$ does in fact equal ${\sf NP}$, how would this enhance our algorithms to factor integers faster. In other words, what kind of insight would this fact give us in understanding integer ...
3
votes
1answer
170 views

Are there are problems in NP that have been shown to be not NP-complete but it is still not known if they are in P or not?

I am not talking about NP-indeterminate class because those problems have to be shown to not exist either in P or NP-complete class and existence of such problems proves P!=NP. I am interested to know ...