Questions tagged [p-vs-np]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
246
votes
7answers
111k 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 ...
96
votes
5answers
14k 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 direction....
54
votes
6answers
21k views

If everyone believes P ≠ NP, why is everyone sceptical of proof attempts for P ≠ NP?

Many seem to believe that $P\ne NP$, but many also believe it to be very unlikely that this will ever be proven. Is there not some inconsistency to this? If you hold that such a proof is unlikely, ...
54
votes
9answers
8k 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 ...
31
votes
4answers
19k 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 ...
28
votes
3answers
2k views

Why is Relativization a barrier?

When I was explaining the Baker-Gill-Solovay proof that there exists an oracle with which we can have, $\mathsf{P} = \mathsf{NP}$, and an oracle with which we can have $\mathsf{P} \neq \mathsf{NP}$ to ...
19
votes
3answers
3k views

Does $\mathsf{P} \ne \mathsf{NP}$ imply that $|\mathsf{NP}| > |\mathsf{P}|$?

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 $\mathsf{P}$ ...
18
votes
5answers
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 ...
14
votes
1answer
2k views

Why do Shaefer's and Mahaney's Theorems not imply P = NP?

I'm sure someone has thought about this before or immediately dismissed it, but why does Schaefer's dichotomy theory along with Mahaney's theorem on sparse sets not imply P = NP ? Here's my ...
14
votes
2answers
2k 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 ...
13
votes
3answers
5k 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 ...
13
votes
1answer
795 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 ...
12
votes
5answers
5k 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 ...
12
votes
2answers
5k views

How to prove P$\neq$NP?

I am aware that this seems a very stupid (or too obvious to state) question. However, I am confused at some point. We can show that P $=$ NP if and only if we can design an algorithm that solves any ...
11
votes
2answers
1k 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 ...
11
votes
1answer
384 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 ...
10
votes
2answers
4k views

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 ...
10
votes
3answers
4k 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. ...
10
votes
1answer
2k views

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 ...
10
votes
3answers
468 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
votes
2answers
11k 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...
9
votes
1answer
331 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 ...
9
votes
0answers
269 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.\ \...
8
votes
2answers
817 views

Is detecting easy instances of NP-hard problems easy?

My question is the following. Assume that $\Pi$ is an NP-hard problem. Given an arbitrary instance $I$ of $\Pi$ and assume that an adversary knows that this instance is easy to solve, is it possible ...
8
votes
2answers
220 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.
8
votes
2answers
743 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 $k>0$ ...
8
votes
1answer
769 views

What is wrong with this conditional proof of P=NP?

I have recently thought up the following proof that L=P implies P=NP. Suppose L=P. Let A be a problem in NP. By the verifier definition of NP, each positive solution to A has a witness that can be ...
8
votes
1answer
225 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 $\mathrm{P}=\...
8
votes
0answers
195 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 ...
7
votes
1answer
896 views

Why the need for TSP solvers when there are SAT solvers?

Concorde TSP is a solver for TSP. SAT solvers are solvers for boolean satisfiability. TSP and SAT are NP-complete. Hence, why spent the time to develop Concorde TSP when there is an abundance of SAT ...
7
votes
1answer
187 views

How to use an old SAT solver to discover a new one, as is done in The Golden Ticket?

In Lance Fortnow's book The Golden Ticket, he mentions that once you have a polynomial-time algorithm for an NP-complete problem, you can use it to find a faster algorithm. Can you tell me how that is ...
7
votes
2answers
717 views

Polytime algorithm for SUBSET-SUM assuming P=NP

In the Wikipedia page on P vs. NP problem there is an algorithm that "solves" SUBSET-SUM in case P=NP in polynomial time. (It's idea is to find a TM that gives a certificate). But it gives "yes" in ...
6
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 ...
6
votes
1answer
709 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 ...
6
votes
1answer
241 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 $\mathsf{P} \neq \mathsf{NP}$. ...
6
votes
1answer
489 views

Why can't we exploite finiteness to prove incompleteness in NP?

It is well established that the class of recursive languages is strictly contained in the class of recursively enumerable languages (Rec $\ne$ RE). Any finite language is decidable and hence can not ...
6
votes
2answers
182 views

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 ...
6
votes
2answers
620 views

What is the utility of proving P=NP if we can't find an algorithm that can solve any NP problem in polynomial time?

Here we see a very interesting attempt to show that $\mathrm{P} \ne \mathrm{NP}$ by Norbert Blum. Here we see 116 previous attempts at solving P vs. NP. Here we see the P vs NP problem defined as: ...
6
votes
1answer
240 views

Is there a philosophical counterpart question to P != NP?

Gödels motivation to prove his incompleteness theorems was the philosophical statement "This sentence is wrong.". Is there a philosophical counterpart to the statement P != NP? For example such ...
5
votes
1answer
2k views

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 ...
5
votes
3answers
689 views

Could an NP-hard problem have a mechanical or physical solution method?

Is there any NP-hard problem that we can find a mechanical "polynomial time" solution to? For example, suppose we construct a graph out of something physical, e.g. we have have pipes through which we ...
5
votes
2answers
352 views

Why does this not prove $P\neq NP$?

Fiorini, Massar, Pokutta, Tiwary and De Wolf (Exponential Lower Bounds for Polytopes in Combinatorial Optimization, Journal of the ACM 62(2):article 17, 2015; PDF, ArXiv) show any linear program that ...
5
votes
1answer
618 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 ...
5
votes
1answer
235 views

Stronger versions of P != NP which better express actual convictions

Does the conviction "L-uniform NC1 != NP is incredibly hard to prove!" express the core of "P != NP is incredibly hard to prove!" in a similar spirit as the conviction "The polynomial hierarchy doesn'...
5
votes
0answers
53 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 ...
5
votes
0answers
120 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
10k 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 ...
4
votes
2answers
351 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 ...
4
votes
2answers
324 views

Is this language depending on P = NP recursive?

Nobody yet knows if ${\sf P}={\sf NP}$. Let us consider the following language $$L = \begin{cases} (0+1)^* & \text{ if ${\sf P}$ = ${\sf NP}$} \\ \emptyset &\text{ otherwise}. \end{cases}$$ ...
4
votes
2answers
1k views

Having problem understanding the formal definition of NP

So I'm having a tad bit of a problem deciphering the formal definition of NP. In my text book (Algorithm Design, Tardos et al) it says that a problem $X$ belongs to $NP$ iff; there exists a "...