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54 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
70 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
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1answer
50 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
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1answer
38 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
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2answers
132 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 ...
0
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2answers
113 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
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0answers
23 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 ...
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0answers
85 views

research on OR and AND compression in SAT formulas

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 ...
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1answer
473 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
82 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, ...
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0answers
67 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
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1answer
184 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 ...
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2answers
136 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
142 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
114 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
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1answer
116 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 ...
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1answer
31 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 ? ...
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1answer
93 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 ...
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1answer
61 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?
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1answer
1k 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
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1answer
994 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
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1answer
98 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, ...
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3answers
2k 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
164 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 ...
2
votes
2answers
200 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
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1answer
86 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 ...
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5answers
17k 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
157 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
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1answer
215 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 ?
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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
869 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
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2answers
118 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 ...
3
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1answer
222 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
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3answers
274 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 ...
9
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2answers
570 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 ...
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1answer
136 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 ...
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2answers
205 views

If the “is P equals to NP?” is a NP-COMPLETE, what does it tell us?. Some conclusions?

If there is someone can prove that the problem "is P equals to NP?" is a NP-COMPLETE problem, what we can conclude from this?
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1answer
134 views

How to show that the set of machines which accept languages in $\mathrm{NP}\smallsetminus\mathrm P$, is decidable only if $\mathrm P=\mathrm{NP}$?

I'd like your help with proving that the language $$L=\{\langle M \rangle \mathrel| L(M) \in \mathrm{NP}\smallsetminus \mathrm{P} \}$$ is decidable iff $\mathrm{P}=\mathrm{NP}$. If ...
8
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2answers
197 views

Can exactly one of NP and co-NP be equal to P?

Maybe I am missing something obvious, but can it be that P = co-NP $\subsetneq$ NP or vice versa? My feeling is that there must be some theorem that rules out this possibility.
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5answers
7k views

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 ...
14
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4answers
3k views

Are there NP problems, not in P and not NP Complete?

Are there any known problems in $\mathsf{NP}$ (and not in $\mathsf{P}$) that aren't $\mathsf{NP}$ Complete? My understanding is that there are no currently known problems where this is the case, but ...