Questions tagged [p-vs-np]

Questions about or related to P vs. NP

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0answers
63 views

Validity of relativization argument

In the book 'Computational complexity' by Arora-Barak, I was studying that any result of TM/ complexity classes that use only the following two properties of TM's are called relativizing results : $I....
-4
votes
0answers
54 views

An instance of Sudoku that can be solved in poly-time, but is it ASP-complete?

Another Solution Problem (ASP) of a problem ƒ is the following problem: for a given instance x of ƒ and a solution s to it, find a solution to x other than s. x = poly-time solvable puzzles s =...
6
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1answer
150 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)?
-2
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1answer
116 views

Does P^NP=NP imply NP=coNP? [closed]

If you have it, the proof would be appreciated. Note: P^NP means P with NP oracle
4
votes
1answer
395 views

Is there a counterexample to this work?

Is there a counterexample to this claim https://arxiv.org/abs/1610.00353? They claim a $O(n^6)$ LP model with simulations to support. I think asking validity is not a reasonable problem. However ...
5
votes
1answer
255 views

Is Murphy's Law of Complexity Theory consistent? What separations/collapses does it imply?

A decade ago I observed what I dub "Murphy's Law of Complexity Theory": whenever a new separation or collapse is discovered, the question is answered in the direction that makes $P\overset?=NP$ most ...
18
votes
1answer
2k views

Algorithm whose running time depends on P vs. NP

Is there a known, explicit example of an algorithm with the property such that if $P\neq NP$ then this algorithm doesn't run in polynomial time and if $P=NP$ then it does run in polynomial time?
15
votes
1answer
613 views

Barriers to show $P=NP$

We all know showing $P\ne NP$ has barriers. We all have studied these barriers because we believe $P\ne NP$. However assume $P=NP$ and there are wise people who believe that possibility exists. If ...
2
votes
1answer
207 views

P vs. NP in a logic with a random oracle

Choose a random oracle $f : \{0,1\}^\ast \to \{0,1\}$, and define the logic $ZFC^f$ by adding a fresh symbol $g$, an axiom that $g$ has the correct type, and one axiom $g(s) = f(s)$ for each $s \in \{...
2
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0answers
120 views

What are some known methods for showing that a class has no complete problems?

The only way that I know of is the way that you can show that $RE \cap coRE$ does not via diagonalization. Mostly curious because if $NP \cap coNP$ has no complete problems then $P \neq NP$. I tried ...
-3
votes
1answer
115 views

What is wrong with this procedure to convert quadratic programming to convex quadratic programming?

Consider the feasibility quadratic program with constraint $$\sum_{i=1}^nc_{i1}x_{i}\leq \ell_1$$ $$\vdots$$ $$\sum_{i=1}^nc_{it}x_{i}\leq \ell_t$$ $$\sum_{i,j=1}^na_{ij}x_{i}x_{j}+\sum_{i=1}^nb_{i}x_{...
3
votes
0answers
208 views

Is there any known strategy that avoids circuits and that respects believed separations to prove $P$ is not $NP$?

Vinay Deolalikar's approach tried to randomness is not strong enough, Blum's proof tried to show $P/poly$ is not strong enough, Mulmuley's and Smale's approach (while not enough to show $P\neq NP$) ...
0
votes
1answer
233 views

${\bf NP} \not = {\bf E}$ and ${\bf PSPACE} \not = {\bf E}$

We know that ${\bf NP} \subseteq {\bf PH} \subseteq {\bf PSPACE}$. We also know that ${\bf E} \subset {\bf EXP}$, where ${\bf E} = \cup_c DTIME[2^{cn}]$ and ${\bf EXP} = \cup_c DTIME[2^{n^c}]$. It ...
2
votes
1answer
479 views

Can one prove the discovery of a P versus NP solution without actually revealing it?

Suppose a person has proved that P≠NP. He wants to let the world know that he has solved the P versus NP problem but does not want to reveal that he has proved P≠NP as opposed to P=NP. Is there any ...
1
vote
1answer
135 views

Unambiguous SAT and sparse languages

What is the consequence if there are only polynomially many 'yes' classes of instances of a language that is polynomial time reducible from a problem equivalent to UnambiguousSAT (such as possibly ...
5
votes
0answers
159 views

Converse to natural proofs theorem?

Natural proofs paper shows 'if there is a natural property not possessed by any function in P/poly then there is no $2^{n^\epsilon}$-hard PRG'. Is it easy to see the converse 'if there is no $2^{n^\...
5
votes
2answers
241 views

ETH-Hardness of $Gap\text-MAX\text-3SAT_{c}$

The PCP theorem can be stated like this : There is a polynomial time reduction from SAT to $Gap\text-MAX\text-3SAT_{c}$ i.e. there is a reduction that maps an instance $\phi$ of SAT to an instance $...
12
votes
3answers
858 views

Why do computer scientists on the whole work under the assumption that P ≠ NP?

Coming from a math background, it seems interesting to me that on the whole computer scientists tend to work under the assumption that $P \neq NP$. While there is no proof either way, generally, ...
16
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0answers
194 views

What is the background in algebraic geometry and representation theory needed for geometric complexity theory? [duplicate]

I'm a mathematics student in my junior year and I'm interested in computational complexity and specially geometric complexity theory. I'm going to learn algebraic geometry and representation theory ...
8
votes
0answers
276 views

What would be the consequences if all _infinite_ NP-complete languages are p-isomorphic?

The famous Isomorphism Conjecture of Berman and Hartmanis says that all $NP$-complete languages are polynomial time isomorphic ($p$-isomorphic) to each other. It has been an early attempt (published ...
2
votes
0answers
111 views

What would faster Fourier Transform(FFT?) and/or multiplication algorithms imply?

There are many problems which have implications on P vs. NP and other complexity classes. Supposing that we're interested in Fourier transforms and/or multiplication algorithms, do faster Fourier ...
6
votes
1answer
470 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 ...
7
votes
1answer
257 views

Two DFA intersection emptiness connections to SETH & L vs P

(re "fine grained complexity") Wehar has proved that Two DFA intersection emptiness in $O(n^{2-\epsilon})$ time → SETH false. does anyone see any particular key proof difficulty, challenge, ...
17
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4answers
2k views

Chaos and the $P{=}NP$ question

I am interested in learning connections between "chaos," or more broadly, dynamical systems, and the $P{=}NP$ question. Here is an example of the type of literature I am seeking: Ercsey-Ravasz, ...
0
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0answers
141 views

Consequences of VP = VNP on randomness

According to the answers in posting it is possible that $\mathsf{VP} = \mathsf{VNP}$ and $\mathsf{P} \neq \mathsf{NP}$ are simultaneously correct. $\mathsf{VP} = \mathsf{VNP}$ implies $\mathsf{P/...
-2
votes
1answer
329 views

What is the status of intermediate problems if P is not NP in worst way imaginable?

Assume $P\neq BPP\neq NP$ with caveat that there is a deterministic algorithm for every $NP$ complete problem with input size $n$ bits in $2^{(\log n)^{1+f(n)}}$ arithmetic operations on $\log n$ ...
4
votes
1answer
209 views

Implications of $\mathsf{P}\neq\mathsf{NP}$ in $\mathsf{BSS}$ model

What are implications of $\mathsf{P}\neq\mathsf{NP}$ in $\mathsf{BSS}$ model to $\mathsf{Turing}$ model and $\mathsf{Valiant's}$ counting complexity model? In opposite direction what are implications ...
12
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1answer
494 views

L/P/PSpace vs P/NP

in 1979 Hopcroft/ Ullman wrote that L ⊆ P ⊆ NP ⊆ PSpace is known but L ⊊ PSpace is the only proper (& trivial) containment known although all are conjectured to be proper containments, and "where ...
-2
votes
1answer
107 views

what can be said about complexity of “typical” supercomputing programs/ applications? any NP hard?

supercomputers have risen dramatically in their computational powers last few decades due to Moore's law & also increasing parallelism technology in hardware and software. many different types of ...
4
votes
1answer
360 views

Analogies between VNP and NP

Valiant introduced the class VNP with respect to "arithmetic circuits" over 35 years ago in a "rough" analogy to NP. Recently, there have been major advances in the area of arithmetic circuits eg as ...
1
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0answers
257 views

What's the Impact if proven NP/co-NP=PSPACE on settling P=NP? The future directions it opens to settle/attack P=NP? Remaining classes left outside?

The primary Impact i know would be that: Polynomial Hierarchy collapses to Level 1. NP=co-NP NP=BPP NP=PSPACE BQP=NP and so on.. What are the attack directions it will open for settling P=NP (in ...
0
votes
1answer
454 views

Is where an oracle B in EXP that $P^B$ $ \ne$ $NP^B$? [duplicate]

According to Baker, Gill, Solovey where is an oracle A in EXP, so $P^A$ = $NP^A$. But is there an oracle B in EXP, what $P^B$ $\ne$ $NP^B$?
1
vote
0answers
264 views

intuition that VP=?VNP is (not?) connected to P=?NP

recently there has been major progress into the VP=?VNP problem for algebraic circuits originated by Valiant, inspiring some optimistic outlook on its eventual or imminent resolution.[1] what is an ...
12
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2answers
487 views

Looking for Literature Source for Following idea

I am quite certain that I am not the first to entertain the idea that I am going to present. However, it would be helpful if I can find any literature related to the idea. The idea is to construct a ...
22
votes
6answers
2k views

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 ...
1
vote
0answers
143 views

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 ...
9
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0answers
480 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. ...
7
votes
2answers
677 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 $...
3
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0answers
242 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 $...
0
votes
1answer
139 views

Assume that $\mathsf{NP} \subseteq \mathsf{P}/\text{log(n)}$, does it imply that $\mathsf{P} = \mathsf{NP}$? [closed]

I am trying to either prove or refute the claim mentioned in the title. Any ideas ?
5
votes
3answers
765 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
vote
1answer
451 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
303 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 ...
18
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1answer
437 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
votes
1answer
228 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 ...
29
votes
6answers
2k 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 ...
12
votes
3answers
740 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 ...
7
votes
1answer
553 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 ...
16
votes
1answer
719 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
votes
2answers
433 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/...