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Questions tagged [p-vs-np]

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238
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6answers
107k 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
13k 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....
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 ...
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 ...
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 ...
11
votes
1answer
376 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 ...
16
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 $\mathsf{P}$ ...
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 ...
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 ...
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 ...
4
votes
1answer
283 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 ...
4
votes
1answer
237 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 ...
1
vote
2answers
1k views

P=NP, isn't it?

Cook and Levin showed in 1971 how deterministically in polynomial time from every non deterministic Turing machine M, that halts in polynomial number of moves/steps, and string w to create the boolean ...
31
votes
4answers
18k 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 ...
8
votes
2answers
716 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 $k>0$ ...
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
459 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 ...
8
votes
2answers
10k 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...
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 ...
2
votes
1answer
521 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 ...
9
votes
1answer
324 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 ...
7
votes
2answers
652 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 ...
3
votes
1answer
5k 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 ...
1
vote
2answers
2k views

If an NP-complete problem is shown to have a non-polynomial lower bound, would that prove that P != NP?

I understand that the Cook-Levin theorem proved that any NP problem is reducible to an NP-complete problem, which signifies that if a polynomial-time algorithm for an NP-complete problem is found, it ...
1
vote
1answer
1k 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, ...
-1
votes
2answers
307 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?