Questions tagged [complexity-classes]

Computational complexity classes and their relations

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2
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0answers
49 views

Given an algebraic variaties of n multivarieties polynomial equations, is there any algorithm to decide whether there is n-cube inscribing to it?

Given an algebraic variaties of n multivarieties polynomial equations, is there any algorithm to decide whether there is n-cube inscribing to it? And if there is, what is the computational ...
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1answer
157 views

What is the computational complexity of solutions over $\mathbb{Q}$ of polynomial equation with coeffiecents over $\mathbb{Z}$

What is the complexity of the following problem? (e.g. best-known running time, space, best upper bound in terms of complexity classes, etc.) Input: A multivariate polynomial $f$ with coefficients ...
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1answer
161 views

Almost-P and related definitions

I'm pretty sure this has a trivial answer but it's always faster to ask the community :-) I understand that, relative to a random oracle, P=BPP. But this is sometimes phrased via the shorthand "...
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1answer
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What is the relation between P-immune languages and NP-complete languages? [closed]

Can a NP-complete language be P-immune? Why can't existence of P-immune languages separate NP from P?
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When studying the computational complexity of functions $\{0, 1\}^\ast \to \{0, 1\}^\ast$, is it enough to restrict to $\{0, 1\}^\ast \to \{0, 1\}$?

I started reading Avi Wigderson's paper $\mathcal{P}$, $\mathcal{NP}$ and Mathematics – a Computational Complexity Perspective (link). (Notation: $\{0, 1\}^\ast$ is the set of all finite binary ...
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1answer
397 views

Is there any known Poly-APX-complete minimimization problem?

All Poly-APX-complete problems I know are maximization problems, e.g. Max Clique, Max Independent Set, Max One for some set of contraints, and even choosing the attributes of a product to maximize (...
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1answer
79 views

Is there an inherently ambiguous language which can not be recognized by Deterministic LBA?

Is there inherently ambiguous language which can not be recognized by Deterministic LBA? For example, $L=\{wv: w,v=(x|y)^*, w=w^R,v=v^R\}$, is there any deterministc LBA that recognizes $L$ ?
2
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1answer
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Is sliding blocks linear space complete?

Sliding blocks is PSPACE complete even in its simplest form involving 1x2 and 2x1 blocks (without rotation or fractional positions) in a rectangular area, with goal being to move a designated block to ...
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2answers
273 views

Example of something that’s different for generic and random oracles?

Let $G$ be a generic oracle in the sense of Cohen / Baire category. Let $R$ be a random oracle. Are there complexity classes A and B with $$\mathrm{A}^G=\mathrm{B}^G\quad\text{and}\quad\mathrm{A}^R\...
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Complexity Class Equalities on the Edge of Inconsistency

What are some of the most extreme potential equalities between computational complexity classes (especially if there is a barrier to refuting them)? These may give us an opportunity to prove better ...
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Is there a language in NSPACE(O(n)) and (very likely) not in DSPACE(O(n))?

Actually I found that the set of context-sensitive Languages, $\mathbf{CSL}$ ($\mathbf{=NSPACE}(O(n))$ $= \mathbf{LBA}$ accepted languages) are not so widely discussed as $\mathbf{REG}$ (regular ...
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2answers
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Is Asymptotic PTAS $\subseteq$ APX?

The definition of asymptotic polynomial-time approximation scheme (Asymptotic PTAS) is defined as follows: A minimization problem $\Pi$ is Asymptotic PTAS if for all $\epsilon$ there exists an ...
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1answer
730 views

Power of randomness vs. power of indefinite computation

I am writing a paragraph on the power of randomness, part of which I am trying to ground in theory of computation (I am no expert/researcher in this field). First off, I am aware that for ...
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3answers
362 views

Classes between $\textbf{PSPACE}$ and $\textbf{EXP}$

1) What classes contain $\textbf{PSPACE}$, are contained in $\textbf{EXP}$ and (presumably) are not equal to $\textbf{PSPACE}$ nor to $\textbf{EXP}$? A possible class satisfying this requirement: the ...
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Logarithmic levels of the polynomial hierarchy (below PSPACE)

We generally define $PH = \cup_i\Sigma_i^p$ (or various equivalent forms.) In the same notation we can also define $PSPACE = \cup_c\Sigma_{n^c}^p$--that is, like the polynomial hierarchy, but with a ...
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2answers
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What are examples of complexity classes that have contradictory relativizations but they were proven to be either equal or unequal?

In this article Chang et al. provide a counterexample by giving an oracle $A$ such that $\mathsf{IP}^A \neq \mathsf{PSPACE}^A$. I wanted to know if there are more examples like this.
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Tardos Function Counterexample to Blum's $P\neq NP$ Claim

In this thread, Norbet Blum's attempted $P \neq NP$ proof is succinctly disproved by noting that the Tardos function is a counterexample to Theorem 6. Theorem 6: Let $f \in \mathcal{B}_n$ be any ...
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1answer
207 views

Cases of Linear programming known to be in $NC$?

Linear programming is $P$-complete. However are there special situations where we know an $NC$ algorithm?
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Is Norbert Blum's 2017 proof that $P \ne NP$ correct?

Norbert Blum recently posted a 38-page proof that $P \ne NP$. Is it correct? Also on topic: where else (on the internet) is its correctness being discussed? Note: the focus of this question text has ...
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2answers
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Oracle comparing $EXP$ with $UP$

Heller (Theorem 6) gave an oracle relative to which $NP=EXP$, and Homer & Selman gave an oracle relative to which $P=UP$ and $\Sigma_2^P=EXP$. Beigel, Buhrman, Fortnow (freely available author's ...
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A succinct version of permanent that is $EXP$-complete

Succinct version of permanent is $NEXP$-hard (https://eccc.weizmann.ac.il/report/2012/086/) and so unlikely to be $EXP$-complete. Permanent mod $2$ is in $\oplus L$ and so succinct version is ...
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variant of Critical SAT

The language Critical SAT is defined as the set of $CNF$ boolean formulas $f$ such that $f \in UNSAT$ but removing any clause from $f$ makes it satisfiable. It is known that Critical SAT is $DP$-...
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Closure properties of $L$ (DLOGSPACE)? [closed]

What are the closure properties of $L$ (DLOGSPACE)? I'm not only intrested in these properties (if of course $S$ and $T$ are in $L$) : $S \cap T$ $S^*$ (kleene-star) $S.T$ (concat)
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1answer
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What does $\#P\subseteq FP^{PPAD}$ imply?

We know $\#P\subseteq {PPAD}\implies PH\subseteq P^{{PPAD}}\subseteq P^{{NP}}$ and the polynomial hierarchy collapses ($FP^{PPAD}=PPAD$ following Emil Jerabek's comment). Can $\#P\subseteq {PPAD}$ ...
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1answer
444 views

Time Hierarchies in DSPACE(O(s(n)))

The time hierarchy theorem states that turing machines can solve more problems if they have (enough) more time. Does it hold in some way if the space is limited asymptotically? How does $\textrm{DTISP}...
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1answer
286 views

Dp completeness of a problem

Given a Boolean formula $\varphi$ over the variables $\{x_1...x_n\}$ , an assignment $T_0$ for $\varphi$ and an integer $k$, I am interested in the following question: Does $k$ is the minimal number ...
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On status of Valiant's $NC^2=P^{\#P}$ provability program?

In here it is written 'A most interesting/controversial talk was by Leslie Valiant. He explored paths to try to prove that $NC^2=P^{\#P}\dots$'.... This was a decade back. What is the rationale (at ...
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1answer
165 views

On $\#P\subseteq FP^{\Sigma_{f(n)}^P}$?

Is it known that permanent of a $0/1$ $n\times n$ matrix $M$ is computable in polynomial or randomized polynomial time with access to a ${\Sigma_{(\log n)^c}^P}$ oracle where $0<c$ holds and $\...
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1answer
131 views

On $\Delta_i^P$

We know $P\subseteq NP\cap coNP\subseteq\Delta_i^P=P^{\Sigma_{i-1}^P}\subseteq \Sigma_i^P\cap\Pi_i^P=NP^{\Sigma_{i-1}^P}\cap coNP^{\Sigma_{i-1}^P}$. If $P=BPP$ is there a 'higher' randomized class ...
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Co-Partition Problem: Why is this proof for it being in NP wrong? [closed]

So I have this wrong proof that the problem "Co-Partition" is in NP. I know the proof is wrong because I've encountered it in an educational environment and was told that it's not working. I don't, ...
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108 views

What is the complexity of computation of zero point by Rieman zeta function?

What is the complexity of computation of zero point by Rieman zeta function? That is, given s, whether ζ(s)=0? Is it in P? Any reference is appreciated.
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1answer
297 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 ...
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1answer
150 views

Descriptive model theory classification of Counting hierarchy

Descriptive model theory uses logic to characterize complexity classes How to model Counting Hierarchy PSPACE in descriptive model theory?
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290 views

Is there a computing problem which is in quasi-polynomial time but is (maybe) not in $\beta P$?

Quasi-polynomial time, or QP for short, is a complexity class on deterministic Turing machine. Here is the precise definition:https://complexityzoo.uwaterloo.ca/Complexity_Zoo:Q#qp While βP is a ...
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1answer
268 views

On $NP$ and $XP$ classes?

On page 33 venn diagram in http://tcs.rwth-aachen.de/~sanchez/slides/Raleigh2014.pdf it is implied that $XP\subseteq NP$. Below this there is a statement which says $XP\not = NP$ unless $P=NP$. Is ...
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Intermediate problems between PSPACE and EXPTIME

Intermediate problems between P and NP are quite famous, and are sometimes considered as complexity classes by themselves. Do you know of any problem that is known to be PSPACE-hard and in EXPTIME, ...
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375 views

Complexity of comparison unary>binary

What is the smallest widely-known complexity class to which $$\left\{\langle i,j\rangle\middle|\begin{array}{@{}l@{\ }l@{}} & i\ \text{is a unary encoding of a positive integer}\ \hat\imath\\\...
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1answer
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Is finding a solution harder than verifying a solution? [closed]

Is there any known problem in Comp science where determinisitically finding a "non-trivial" solution to that problem is asymptotically easier than verifying a solution?
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1answer
389 views

A question of relationships between #P and PSPACE [closed]

Let us assume there is some machine X that converts boolean formula to following form in polynomial time: $$\Phi(x_1, x_2 .. x_m) = r_1(x_{i_1}, x_{j_1}, x_{k_1}) \land r_2(x_{i_2}, x_{j_2}, x_{k_2}) ...
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1answer
295 views

Does there exist an oracle $A$ such that $(P^{\#P})^{A} \neq PSPACE^{A}$?

Background We know that $P^{\#P} \subseteq PSPACE$. In addition, we known from Toda's theorem that $PH \subseteq P^{\#P}$. For more background on $\#P$, see here: https://en.wikipedia....
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Does the existence of an RP-complete language imply P = RP?

(I'm not sure if this is research-level, but I couldn't find an answer to this question elsewhere) The question of whether there exists an RP-complete language seems to be open, but I guess we believe ...
3
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1answer
253 views

Looking for approximation class between NPO and Exp-APX

I'm trying to identify the approximation hardness of some maximization problem A. In problem A, finding a solution whose quality is 0 (i.e. such that the value returned by the objetive function is 0) ...
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1answer
85 views

Running multiple rounds of a BQP computation, without multiple measurements? [closed]

BQP as usually defined is: the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances. Just like BPP, the choice of 1/...
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1answer
642 views

Is $\sf{P^{NP \cap coNP}} = \sf{NP \cap coNP}$?

If it is unknown, are there reasons to believe that they might not be equal?
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1answer
346 views

What's the relationship between ASP-complete and #P-complete?

Given that ASP-reductions by definition are parsimonious and parsimonious reductions preserve #P-completeness, one might think that the counting version of all ASP-complete problems are also #P-...
6
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1answer
2k views

k-Vertex Cover problem is in parameterized Log space

$k$-Vertex Cover: Given a graph $G = (V, E)$ where $V$ is a set of vertices and $E$ a set of edges, and an integer $k$, the $k$-Vertex Cover problem determines if there exists a subset of vertices $...
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Non-trivial PCP characterizations of complexity classes beyond ELEMENTARY?

There are interesting results of the form $PCP[a(n), b(n)] = \texttt{SOMECLASS(n)}$ for multiple classes in the exponential hierarchy: the most famous one is probably $PCP[O(log(n)), O(1)] = NP$. Are ...
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1answer
147 views

Finding a certificate if E=NE

If $\textbf{P}= \textbf{NP}$ then for every language from $\textbf{NP}$ there exists an algorithm of finding a certificate in polynomial time. Assume that $\textbf{E} = \textbf{NE}$. Is it true that ...
4
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1answer
149 views

$NotTooManyP^{cc}$ class in communication complexity

Class $P^{cc}$ is class of languages admitting deterministic communication protocol with polylog bits of communication. Class $NP^{cc}$ is class of languages admitting nondeterministic communication ...
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119 views

What can we say about AM[log n]?

It is known that $\textbf{AM}[O(1)] = \textbf{AM}$. Since $\textbf{IP}=\textbf{PSPACE}$ we have $\textbf{AM}[poly(n)] = \textbf{PSPACE}$. Can we say something about $\textbf{AM}[ f(n)]$, where $f$ ...

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