Questions tagged [p-vs-np]

Questions about or related to P vs. NP

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Proofs, Barriers and P vs NP

It is well known that any proof resolving the P vs NP question must overcome relativization, natural proofs and algebrization barriers. The following diagram partitions the "proof space" into ...
Shiva Kintali's user avatar
56 votes
9 answers
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Explain P = NP problem to 10 year old

It is my first question on this site. I am taking a master's course on theory of computation. How you would explain P = NP problem to a 10 year old child and why it has such a monetary reward on it? ...
39 votes
2 answers
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Mulmuley's GCT program

It is sometimes claimed that Ketan Mulmuley's Geometric Complexity Theory is the only plausible program for settling the open questions of complexity theory like P vs. NP question. There has been ...
Anonymous's user avatar
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29 votes
4 answers
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Implications of unprovability of $P\neq NP$

I was reading "Is P Versus NP Formally Independent?" but I got puzzled. It is widely believed in complexity theory that $\mathsf{P} \neq \mathsf{NP}$. My question is about what if this is ...
karthik's user avatar
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19 votes
1 answer
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Natural candidate against the Isomorphism Conjecture?

The famous Isomorphism Conjecture of Berman and Hartmanis says that all $NP$-complete languages are polynomial time isomorphic (p-isomorphic) to each other. The key significance of the conjecture is ...
Andras Farago's user avatar
7 votes
2 answers
2k views

P vs. NP and Pseudorandom Bit Generators

According to an article on pseudorandom number generators (PRNG) by Jeff Lagarias, he states that trying to prove that a PRNG is unpredictable (secure) is just "as hard" as trying to prove ...
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14 votes
3 answers
1k views

What are natural examples of non-relativizable proofs?

As I understand it, a proof that P=NP or P≠NP would need to be non-relativizable (as in recursion theory oracles). Virtually all proofs seem to be relativizable, though. What are good examples of ...
Sai's user avatar
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8 votes
2 answers
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Baker Gill Solovay $P^B \ne NP^B$ relativization, what class is $B$ in?

A recent question asks whether relativization is well-defined. This question wonders how to put one use of it on firmer ground. In the BGS 1975 proof that there exists a language $B$ such that $...
vzn's user avatar
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6 answers
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Why are so few natural candidates for NP-intermediate status?

It is well known by Ladner's Theorem that if ${\mathsf P}\neq \mathsf {NP}$, then there exist infinitely many $\mathsf {NP}$-intermediate ($\mathsf{NPI}$) problems. There are also natural candidates ...
Andras Farago's user avatar
29 votes
3 answers
1k views

Is NPI contained in P/poly?

It is conjectured that $\mathsf{NP} \nsubseteq \mathsf{P}/\text{poly}$ since the converse would imply $\mathsf{PH} = \Sigma_2$. Ladner's theorem establishes that if $\mathsf{P} \ne \mathsf{NP}$ then $\...
Vanessa's user avatar
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25 votes
4 answers
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Best known deterministic time complexity lower bound for a natural problem in NP

This answer to Major unsolved problems in theoretical computer science? question states that it is open if a particular problem in NP requires $\Omega(n^2)$ time. Looking at the comments under answer ...
Anonymous's user avatar
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24 votes
6 answers
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Statements that imply $\mathbf{P}\neq \mathbf{NP}$

This is sort of an open-ended question - for which I apologize in advance. Are there examples of statements that (seemingly) don't have anything to do with complexity or Turing machines but the ...
Dominic van der Zypen's user avatar
18 votes
4 answers
2k views

List of theorems stating that P does not equal NP if and only if

I think it would be a good idea to make a list of theorems stating that P does not equal NP if and only if such and such exits, some complexity class is contained in another complexity class and so on ...
Tayfun Pay's user avatar
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12 votes
2 answers
1k views

Reducing P vs. NP to SAT

The following question uses ideas from cryptography applied to complexity theory. That said, it is a purely complexity-theoretic question, and no crypto knowledge whatsoever is required to answer it. ...
Sadeq Dousti's user avatar
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9 votes
3 answers
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Could there be an extremely large hidden subset of Polynomially solvable problems within NP-Complete problems?

Suppose P != NP. We know that we can make easy instances of 3-SAT at any time. We can also generate what we believe to be hard instances (because our algorithms can't solve them quickly). Is there ...
Elliot JJ's user avatar
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8 votes
1 answer
429 views

Is $\mathsf{P} = \mathsf{NP}$ relative to a universal predictor?

Consider any language $L$. Define $s(L) \in {\lbrace 0, 1 \rbrace}^\omega$ (an infinite sequence of bits) by the recursive formula $$s(L)_n=\chi_L(s(L)_{<n})$$ Here $\chi_L$ is the characteristic ...
Vanessa's user avatar
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8 votes
1 answer
317 views

Is there a relation between BBH (black box hypothesis) and SETH (strong exponential time hypothesis)?

Is there a relation between BBH (black box hypothesis) and SETH (strong exponential time hypothesis)?
user52643's user avatar
6 votes
1 answer
733 views

Is "two or zero" matching in a bipartite graph NP complete?

I have this problem, which I haven't seen before in the literature: given a bipartite graph and a natural number $k$, can we select at least $k$ of the edges such that each left-hand vertex is ...
James Trimble's user avatar
5 votes
0 answers
242 views

Non-uniform average-case complexity of NP

It is conjectured that NP-complete problems are hard not only in the worst case but also in the typical case. Formally, given a language $S \in \lbrace 0,1 \rbrace^*$ and for each $n$ a probability ...
Vanessa's user avatar
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4 votes
2 answers
553 views

P/poly vs NP separation based on circuit trees instead of DAGs

there are various theorems that relate major complexity class separations to circuit family DAGs sizes, in particular for P/poly vs NP. in contrast, are there theorems/conjectures that relate P/...
vzn's user avatar
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0 votes
0 answers
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P vs. NP via psuedo-random number generators [duplicate]

Possible Duplicate: P vs. NP and Pseudorandom Bit Generators Hello again , and thank you all for making this website a great vehicle for knowledge exchange. So my question is , are you trying to ...
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-5 votes
1 answer
277 views

Can on every instance P = NP? [closed]

I want to ask a question concerning some aspects of the P vs. NP problem. There are some results that people use to support a preference / belief about the conjecture $P \neq NP$ (e.g. separation of P ...
Nikos M.'s user avatar
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