Computational complexity classes and their relations

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Proof that BQP is contained in this classical complexity class [on hold]

I am trying to show that BQP is contained in the the complexity class class of decision problems for which there exists a classical algorithm that, for problem size n, solves the problem using an ...
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$\mathsf{EXP}$ vs $\oplus\mathsf{EXP}$

In our recent work, we resolve a computational problem which arose in combinatorial context, under assumption that $\mathsf{EXP} \ne \mathsf{\oplus{}EXP}$, where $\mathsf{\oplus{}EXP}$ is the ...
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Example of a function problem which is $\mathrm{FP}^{\mathrm{NP}}[wit, log]$-hard?

The usage of an $\mathrm{NP}$-oracles which delivers a witness has been proposed for example in [Buss1995]. I would like to see an example of an $\mathrm{FP}^{\mathrm{NP}}[wit, log]$-hard problem. Can ...
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What's wrong here, or, is CNF to DNF conversion in o(exp(n))? [migrated]

I've been thinking about conversion from CNF to DNF. Assume a "worst case" CNF formula with $k$ disjunctions, each containing exactly $l$ elements and no variable is used twice. Example with $k=3$ and ...
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complexity class of problems [on hold]

What does complexity class $NP^{{(\Sigma^p_2})^c}$ mean? Can you please give an example of a decision problem that is solvable in this class?
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Computational tractability

Consider the following optimization problem: $\max_{x\in X}\min_{y\in Y} ||f(x,y)||_2$ $X$ and $Y$ are given $H$-polytopes in the positive orthant of $\mathbb{R}^n$ and $f(x,y)$ is a biconvex ...
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Complexity of max problem

Consider the problem $\max_x \;||x||_2\\ x\in P\subseteq \mathbb{R}_{\geq 0}^n$ where $||\cdot||$ is Euclidean 2-norm and $P$ is a polytope in positive orthant of $\mathbb{R}^n$. Is this problem ...
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Is this some variant of the Knapsack Problem?

We are a set of items $I = \{I_1, I_2,.. I_n\}$, which need to be placed in a certain number of knapsacks $K$. We can use as many knapsacks as we want and each knapsack has an infinite capacity but ...
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Problems in BQP but conjectured to be outside P

Wikipedia listed four problems that are in $BQP$ but conjectured to be outside $P$: Integer factorization; Discrete logarithm; Simulation of quantum systems; Computing the Jones polynomial at certain ...
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#P- vs PP-Completeness

Suppose $A$ is any #P-complete problem. Now, $A$ is modified to obtain a decision problem $A'$ not by asking whether there is a solution but whether at least half of the potential solutions are ...
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Easy decision hard counting Parametrized

It is known that counting perfect matchings in a bipartite graph is #P-complete. On the other hand, finding a perfect matching belongs in P. Is there a problem, that exhibits the same behavior in ...
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About the sparsest-cut question

Can someone kindly help clarify as to exactly what is the generally accepted definition of the "sparsest cut" problem for a graph? (Isn't the set which achieves the Cheeger constant for a graph, ...
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Impact of proof of NP=co-NP on RP vs co-RP Question?

It is known that P ⊆ RP ⊆ NP and P ⊆ co-RP ⊆ co-NP. In an oracle world: If NP=co-NP, does RP=co-RP=ZPP follow automatically or does it require additional conditions? If NP=PSPACE, does RP=co-RP=ZPP ...
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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 ...
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To which complexity class does this language belong?

I was thinking about which class this language belongs: $L =\{ \langle G,k \rangle \mid G $ is a graph, $k$ is a natural number and $k$ is the chromatic number of $G\}$ I thought of $L$ as (1) " ...
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Can differential equations be classed into their own complexity classes?

Problems have been, as a whole, classified, thanks to Computational Complexity. But, in differential equations, is it possible to classify differential equations depending on their computational ...
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Problems with big open complexity gaps

This question is about problems for which there is a big open complexity gap between known lower bound and upper bound, but not because of open problems on complexity classes themselves. To be more ...
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possible bridge between group growth theory and complexity theory?

RJ Lipton conjectures a link between group growth theory and complexity theory. Group growth theory has undergone rapid advance in the last decade and has many surface similarities/ parallels with ...
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If P = NP were true, would quantum computers be useful?

Suppose that P = NP is true. Would there then be any practical application to building a quantum computer such as solving certain problems faster, or would any such improvement be irrelevant based on ...
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Is there a PPAD algorithm for computing primes that sum to even numbers?

Goldbach's conjecture states that every even number greater than 2 can be expressed as the sum of 2 primes. I'm interested in this function problem: Given an even natural number n greater than 2, ...
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Complexity of coloring in weakly perfect graphs?

A graph is weakly perfect if the clique number equals the chromatic number, i.e. $\omega(G)=\chi(G)$. Deciding membership is NP-complete according to the paper. Because of the inequality $\omega(G) ...
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Consequences of the existence of the following algorithm: does it imply any complexity class separation / collapse?

Let $G$ be a $3$-regular graph. Let $O$ be the number of vertex covers of $G$ having odd cardinality, and let $E$ be the number of vertex covers of $G$ having even cardinality. Let $\Delta = O - E$. ...
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Problems with exponential lower computational complexity bound [closed]

I am looking for problems which have an algorithm with asymptotically optimal exponential computational complexity or the problem has a lower bound of exponential computational complexity for ...
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3answers
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How hard is it to find a “well-distributed” subset of models of a propositional formula?

We consider the propositional language $\mathcal{L}_{\mathit{PS}}$ defined over a finite alphabet $\mathit{PS}$ and the usual logical connectives. An interpretation is an assignment $\mathit{PS} ...
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Can typed lambda calculi express *all* algorithms below a given complexity?

I know that the complexity of most varieties of typed lambda calculi without the Y combinator primitive is bounded, i.e. only functions of bounded complexity can be expressed, with the bound becoming ...
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Randomized Algorithm with random input

As per the Yao's principle, in order to lower bound the cost of Randomized algorithm it suffices to find an appropriate distribution of difficult inputs, and to prove that no deterministic algorithm ...
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Understanding MA protocol as a variant of TM for small space setting

MA protocol is one of the most basic models of interactive proofs. Merlin is a prover sending a witness $w$ for given input string $x$, and Arthur is a verifier who verifies if $w$ is a positive ...
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What is the major difference between PP and RP? [closed]

So according to complexity zoo, the definition of RP is: The class of decision problems solvable by an NP machine such that 1.If the answer is 'yes,' at least 1/2 of computation paths accept. ...
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Determine the complexity class of a language [closed]

Let $L'$ be the language containing all the pairs $(G,v)$ where $G$ is a directed graph and $v$ is a vertex in $G$ such that $G$ contains a cycle (i.e. closed walk) that contains $v$ and the number of ...
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Minimum Tree-width of circuit for MAJORITY

What is the minimum tree-width of a circuit over $\{\wedge,\vee,\neg\}$ for computing MAJ? Here MAJ $:\{0,1\}^n \rightarrow \{0,1\}$ outputs 1 iff at least half of its inputs are $1$. I care ...
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What if an $\mathsf L$-complete problem has $\mathsf{NC}^1$ circuits? More generally, what evidence is there against $\mathsf{NC}^1=\mathsf{L}$?

Edit: let me reformulate the question in a more specific way (and change the title accordingly). A slightly edited version of the original question follows. Is there a result comparable to the ...
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Is P equal to the intersection of all superpolynomial time classes?

Let us call a function $f(n)$ superpolynomial if $\lim_{n\rightarrow\infty} n^c/f(n)=0$ holds for every $c>0$. It is clear that for any language $L\in {\mathsf P}$ it holds that $L\in {\mathsf ...
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Consequences of $\oplus \mathbf{P} \subseteq \mathbf{NP}$

I have part of a proof attempt of $\oplus \mathbf{P} \subseteq \mathbf{NP}$. The proof attempt consists of a Karp reduction from the $\oplus \mathbf{P}$-complete problem $\oplus$3-REGULAR VERTEX COVER ...
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Is there a reason we haven't been able to prove that the existence of natural NPI problems even conditionally under assumption NPI is not empty?

We can write ${\mathsf {NP}}-{\mathsf P}= {\mathsf {NPC}}\cup {\mathsf {NPI}}$ where ${\mathsf {NPC}}$ is the set of ${\mathsf {NP}}$-complete languages (not in ${\mathsf {P}}$ by this partition), ...
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#P-complete problems are at least as hard as NP-complete problems

I just read J. Scott Provan, Michael O. Ball: The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected. SIAM J. Comput. 12(4): 777-788 (1983) and one of the first ...
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Large classes which contain LOGSPACE for which strict inclusions are unknown

The wikipedia page on PSPACE mentions that the inclusion $NL\subset PH$ is not known to be strict (unfortunately without references). Q1: What about $L\subset PH$ and $L\subset P^{\#P}$ - are these ...
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proving speedup phenomenon does not apply to any open complexity class separations

Aaronson recently wrote a blog refuting the idea that there could be some "glitch" in the formulation of the P vs NP conjecture[1] which reminds me of this following question. the Blum speedup ...
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How powerful are weak complexity classes with powerful oracles?

I am interested in complexity classes of the form $A^{B}$, where $A$ and $B$ are complexity classes such that $A \subsetneq \mathsf{P}$ and $\mathsf{NP} \subsetneq B$ are (believed to be) true. ...
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Factoring as a decision problem

I've seen in multiple places stating that factoring is in BQP and referencing Shor's algorithm, but Shor's algorithm is not solving a decision problem. How can factoring be restated in a decision ...
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Example of #P-intermediate problem

The previous question Do there exist intermediate problems (in the sense of Ladner's Theorem) for FP vs. #P? I assume that something is known, because I read some papers concerned with FP/#P ...
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Does $\# \mathsf{P}\subseteq \mathsf{FP}^{\mathsf{PH}}$?

The Toda's theorem is a relationship between two different complexity classes: $ \# \mathsf{P} $ and $PH$. He proved that $ \mathsf{PH}\subseteq \mathsf{P}^{\#\mathsf{P}} $. I wonder the following ...
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Padding Arguments for Probabilistic Classes

Do padding arguments exist for probabilistic classes? For example, would $P=BPP\Rightarrow EXP=BPEXP$? What about for space bounded computation? Would constant space derandomization imply $L=RL$ or ...
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Known time complexity advantage of quantum algorithms over classical algorithms [duplicate]

I know that this question may depend on how one formulates each complexity class, but in general, what time complexity advantage does quantum algorithms have over classical algorithms?
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NP-complete problem with polynomially many certificates?

Let's call a language $L \in$ NP sparsely certificated if and only if: There exists a polynomial $p : \mathbb{N} \rightarrow \mathbb{N}$ such that for every input $x \in \Sigma^*$ of size $n$, if $x ...
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Problems in $\text{PSPACE} \cap \text{Co-NP-Hard}$

I'm in search for examples of decision problems lying in $\text{PSPACE}\cap \text{coNP-hard}$ which are also not (known to be) in $\text{coNP}\cup \text{NP}\cup \text{NP-hard}\cup ...
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Is there an oracle separating Parity-P from PSPACE?

Is $ (\oplus \mathsf{P})^A \not \supseteq \mathsf {PSPACE}^A$ for some language $A$?
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Maximizing a #P-hard function

Suppose I have a #P-hard function $f(S,x)$ where $x\in T$. Is the problem of $\arg\max_x f(S,x)$ guaranteed to be intractable? If so, I want to see some references on this topic. If not, is there a ...
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Problems in AM or in MA

What are the examples of problems known to be in $\mathsf{AM}$ (resp. $\mathsf{MA}$) which are not known to be in $\mathsf{NP}$ nor in $\mathsf{BPP}$? For $\mathsf{AM}$, I know the following two ...
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Examples of $\Sigma_2^p$ complete problems?

I need a list of $\Sigma_2^p$ complete languages. There are two such problems listed in the Complexity Zoo, namely: Minimum equivalent DNF. Given a DNF formula F and integer k, is there a DNF ...
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Complexity class of a subset problem similar to subset sum

What could be the complexity class of the following problem: Given a positive integer $K$, a positive integer $d$ (say $2$) and a set $S$ of all non-negative integers less than $K$ find a $S' ...