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0answers
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What is known about lower bounds on the algorithm if P=NP

I recall Scott Aaronson making the claim that either $P\neq NP$ or he has superpowers. What reason is there, if any, to believe that it is not the case that any decision procedure has constants (or ...
7
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0answers
294 views

Assigning probability to membership in an NP-complete language

Motivation Assuming $\mathsf{P}\ne\mathsf{NP}$, it is impossible to efficiently decide membership in an NP-complete language. I would like to assign probability to such membership, in some sense. ...
5
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2answers
261 views

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 ...
2
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0answers
173 views

Does $P\neq NP$ imply any larger separation?

I've asked a similar question in cs.se, but didn't get a satisfying answer. Assuming $P\neq NP$, what can we say about the runtime of any algorithm for an $NP$-complete problem? Obviously, it means ...
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1answer
95 views
5
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3answers
351 views

If SAT is in PCP, for some constant q, then P = NP

I have seen this statement before, but I haven't really seen a proof of it: If $SAT\in PCP_{1,2^{−q}}[\log(n),q]$, for some constant $q$, then $P = NP$. Now, if $SAT\in PCP_{1,2^{−q}}[\log(n),q]$, ...
-1
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1answer
176 views

equivalent way(s) of expressing P=?NP problem in linear programming?

the paper "In defense of the Simplex Algorithm's worst-case behavior" Disser/Skutella [1] was recently cited on this tcs.se site by saeed on another interesting question. the paper introduces the idea ...
6
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2answers
220 views

NPI-candidate hereditary graph property?

A graph property is called hereditary if it is closed with respect to deleting vertices. There are many interesting hereditary graph properties. Moreover, a number of nontrivial general facts are ...
15
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1answer
250 views

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 ...
-5
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1answer
124 views

Can on every instance P = NP? [closed]

I want to ask a question concerning some aspects of the P vs. NP problem. NOTE: Possibly by "popular" standards i am a crank (i tend towards P=NP), but lets focus on the issue (please with a grain of ...
23
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6answers
800 views

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 ...
9
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3answers
427 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 ...
5
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2answers
313 views

Public-key encryption without the assumption that $P \neq NP$

I'm not talking about the RSA, El-gamal, nor any specific encryption scheme. Rather, my question, as related to this and this threads, is why the idea of Public-Key encryption scheme cannot be secure ...
14
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1answer
524 views

Gowers “discretized Borel determinacy” approach

Gowers has recently outlined a problem, which he calls "discretized Borel determinacy," whose solution is related to proving circuit lower bounds. Can you provide a summary of the approach that is ...
4
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2answers
263 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 ...
5
votes
2answers
516 views

On the provability of P versus NP

First of all, my understanding on Gödel's incompleteness theorem (and formal logic in general) is very naive, also is my knowledge on theoretical computer science (meaning only one graduate course ...
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3answers
449 views

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 ...
21
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4answers
1k views

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 ...
4
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2answers
573 views

P vs NP: Instructive example of when Brute Force search can be avoided

To be able to explain the P vs NP problem to non-mathematicians I would like to have a pedagogical example of when Brute Force-search can be avoided. The problem should ideally be immediately ...
0
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0answers
213 views

What are the consequences of a ${\bf O}$(m) algorithm for SAT

We are given a Boolean formula $F$ in conjunctive normal form with $n$ variables and $m$ clauses and we would like to know if there exists at least one assignment to the $n$ variables that makes $F$ ...
8
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1answer
287 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 ...
25
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7answers
5k views

Should we consider $\mathsf{P} \neq \mathsf{NP}$ a law of nature?

Many experts believe that the $\mathsf{P} \neq \mathsf{NP}$ conjecture is true and use it in their results. My concern is that the complexity strongly depends on the $\mathsf{P} \neq \mathsf{NP}$ ...
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0answers
268 views

$\mathsf{P} \ne \mathsf{P/poly} \cap \mathsf{NP}$? [duplicate]

Assuming $\mathsf{P} \ne \mathsf{NP}$ can we show $\mathsf{P} \ne \mathsf{P/poly} \cap \mathsf{NP}$? Obviously this would be the case if $\mathsf{P} \ne \mathsf{NP}$ and $\mathsf{P/poly} \supset ...
9
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0answers
194 views

Are there sampNP-intermediate problems?

I approximately copied the brief "introduction" to average-case complexity theory of NP from my previous question. However, this question is completely different, so please read on It is conjectured ...
4
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0answers
128 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 ...
12
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1answer
331 views

Optimal NP solvers

Fix $X \subset \lbrace 0,1 \rbrace^* \times \lbrace 0,1 \rbrace^*$ an NP-complete search problem e.g. the search form of SAT. Levin search provides an algorithm $L$ for solving $X$ which is optimal in ...
26
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3answers
705 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 ...
11
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3answers
492 views

Recent publications on NP ?= coNP question

I am interested in the question of whether NP is equal to coNP or not. I'd much appreciate some advice on good publications to read on the topic. For the record, I know that this question is ...
5
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1answer
298 views

Do we know that the P vs. NP question isn't affected by Gödels incompleteness theorem? [duplicate]

Possible Duplicate: Implications of unprovability of $P\neq NP$ I shortly came across Gödels incompleteness theorem again and I wondered, since so much time has been spent on trying to ...
4
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0answers
482 views

Does L=P imply any new complexity class separations?

If L=P then P is not equal to PSPACE. This follows from PSPACE properly containing L. I am wondering if L=P implies any stronger separation between complexity classes? Does it imply P is properly ...
1
vote
1answer
321 views

Statement that inequality of P and NP obstructs finding a proof of their separation

In this set of video lectures, Prof. Mulmuley emphasizes several times that P vs. NP is a very deep philosophical question at the core of Mathematics and that the fact that the two classes are not ...
11
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4answers
1k views

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 not ...
26
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1answer
832 views

Toy examples for barriers to $P \ne NP$

Are there any toy examples that provide 'essential' insights into understanding the three known barriers to $P = NP$ problem - relativization, natural proofs and algebrization?
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2answers
1k views

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 ...
5
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0answers
595 views

What is the status of Vladimir Romanov's proof that P=NP? [closed]

A couple of months ago there was a minor hubbub about a paper published on ArXiV by V. F. Romanov titled Non-Orthodox Combinatorial Models Based on Discordant Structures which claimed to give a ...
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9answers
13k views

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? ...
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10answers
932 views

Resources to learn about the P vs. NP problem

I was recently reminded about the $\mathsf{P}$ vs. $\mathsf{NP}$ problem as explained by Stephen A. Cook on Clay Mathematics Institute. It has piqued my interest and I would like to learn more about ...
0
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0answers
177 views

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 ...
6
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2answers
709 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 that P!=NP. ...
11
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2answers
712 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. ...
19
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3answers
838 views

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 ...