Computer Science Stack Exchange
Computer Science Stack Exchange

Questions tagged [p-vs-np]

295 questions
336
votes
7 answers

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 what they actually mean. Wikipedia isn't much help…
Mirrana
  • 4,419
  • 6
  • 22
  • 22
109
votes
5 answers

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. I know that there are approaches that have been…
Raphael
  • 73,212
  • 30
  • 182
  • 400
69
votes
9 answers

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 within the realm of computer science. My question…
RLH
  • 869
  • 1
  • 8
  • 10
64
votes
6 answers

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, then you should also believe that sound arguments…
pafnuti
  • 739
  • 1
  • 5
  • 7
47
votes
4 answers

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 it hasn't been ruled out as a possibility. If…
vpiTriumph
  • 593
  • 1
  • 4
  • 7
37
votes
3 answers

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 a friend, a question came up as to why such…
Nikhil
  • 639
  • 6
  • 12
24
votes
2 answers

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...
Mirrana
  • 4,419
  • 6
  • 22
  • 22
24
votes
5 answers

P = NP clarification

Let's use Traveling Salesman as the example, unless you think there's a simpler, more understable example. My understanding of P=NP question is that, given the optimal solution of a difficult problem, it's easy to check the answer, but very…
Tom Mercer
  • 351
  • 2
  • 6
23
votes
3 answers

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 $\mathsf{P}$ and $\mathsf{NP}$ must have different…
Jason Baker
  • 417
  • 3
  • 5
22
votes
4 answers

How hard would it be to state P vs. NP in a proof assistant?

GJ Woeginger lists 116 invalid proofs of P vs. NP problem. Scott Aaronson published "Eight Signs A Claimed P≠NP Proof Is Wrong" to reduce hype each time someone attempts to settle P vs. NP. Some researchers even refuse to proof-read papers settling…
Isinlor
  • 335
  • 2
  • 6
19
votes
7 answers

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 that A correctly solves Q. To prove P$\neq$NP, we…
Kaalouss
  • 477
  • 1
  • 4
  • 7
16
votes
2 answers

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 factorization better?
Mike G
  • 461
  • 3
  • 14
14
votes
3 answers

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 given instance of problem in NP in polynomial…
padawan
  • 1,455
  • 1
  • 12
  • 30
14
votes
1 answer

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 reasoning: Create a language $L$ which is equal to SAT…
Ari
  • 1,661
  • 10
  • 23
14
votes
5 answers

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 decider for A. Let B be the complement, i.e. B is in…
simpleton
  • 171
  • 1
  • 5
1
2 3
…
19 20