Questions tagged [complexity-classes]
Computational complexity classes and their relations
618
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How can I calculate the computational complexity of an equation composed of 2n multiplications and 2nm^2 additions? [closed]
I want to calculate the computational complexity in term of the big (O).
My equation is:
It composed of 2n multiplications and 2nm^2 additions.
The complexity of this equation is it O( 2n + 2nm^2 ) ...
- 93
0
votes
0
answers
66
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Simplify or bound using Big-O notation
I was following a research paper which have the following equation:
$\left(1-\frac{1}{K}\right)^{K-i}\left[1-\left(1-\frac{1-p}{K}\right)^{i}\right]=\frac{i(1-p)}{K}+O\left(\left(\frac{i}{K}\right)^{2}...
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2
votes
1
answer
126
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Complexity of the Complete (3,2) SAT problem?
A complete $k$-CNF formula is a $k$-CNF formula which contains all clauses of size $k$ or lower it implies.
Deciding the satisfiability of a $k$-CNF formula is clearly a tractable problem since a $k$-...
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34
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Is there an MMSNP formula for 3-colouring?
By MMSNP, I mean Monotone Monadic SNP without inequality. For $k\in\mathbb{N}$, the problem $k$-COLOURABILITY takes a graph $G$ as input and asks whether $G$ is $k$-colourable.
It is well-known that ...
- 1,603
2
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75
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Nondeterministic polynomial time languages with linearly bounded certificates
Define the class $X$ of languages by the condition that a language $L$ over alphabet $\Sigma$ is in $X$ iff there are a constant $c > 0$ and a polynomial-time checking relation $R$ such that for ...
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0
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76
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Does NP-completeness in one graph class imply not NP-intermediate in another graph class?
I am trying to wrap my head around implications of CSP dichotomy theroem.
CSP is short for Constraint Satisfaction Problem.
The following seem to be known results (I shall focus on decision problems ...
- 1,603
6
votes
1
answer
150
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Reductions weaker than polynomial-time for $\exists \mathbb{R}$
I am currently studying the complexity class $\exists \mathbb{R}$ which contains all problems that are reducible in polynomial time to the existential theory of the reals. In the literature ...
- 161
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0
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78
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What are the capabilities of current Boolean Satisfiability Solvers?
I am wondering how well current Boolean Satisfiability solvers are able to perform — for example, are the best SATsolvers able to solve problems from SATlib?
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3
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66
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Complete problems for fast-growing hierarchy classes
I need examples of natural complete problems in classes $\textbf{F}_\alpha$, definition of $\textbf{F}_\alpha$ can be found here. Also in section 6 there are examples for $\omega$, $\omega^\omega$, $\...
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1
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Question about BPP complexity class [closed]
Good morning everyone, I just started studying the BPP complexity class and the amplification lemma. There is one exercise about BPP that I don't understand, I hope that you can help me.
Let $L$ be a ...
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0
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69
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APX-hardness of Max-3-SAT(5)
Max-3-SAT($k$) for a natural number $k$ is the task of finding the maximum number of satisfiable clauses in a Boolean formula in CNF, where every clause contains at most 3 literals and every variable ...
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0
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1
answer
178
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Is there a precise definition for big O notation with 2 or more variables [closed]
Big O notation has a very precise definition for 1 variable. You can prove O(2x^2) = O(x^2) for example. There is never ambiguity. However, for 2 or more variables ...
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6
votes
1
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202
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Complexity of reachability in directed rooted forests
I'm trying to figure out the complexity of the reachability problem having as input a directed rooted forest, i.e., given a set of directed rooted trees and two vertices $s$ and $t$, tell if $s$ and $...
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6
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1
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What is the complexity of HORN-2CNF entailment?
I know the entailment of a propositional variable in a HORN-3CNF formula is $P$-complete.
I can't find any publication in which it has been shown the complexity of the same problem for HORN-2CNF ...
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0
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93
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Query about P/poly and Polynomial Hierarchy Collapse to $\Sigma _{2}$
I am not conversant in the complexity class $P/poly$. While reading about the class on wiki I encountered two conditional statements about it, namely:
If $NP ⊆ P/poly$ then $PH$ (the polynomial ...
- 801
8
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1
answer
318
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Complete problem in $\Sigma_2^p$ - $\Sigma_{2}SAT$
It is known that the following problem is complete in $\Sigma_2^p$:
$\Sigma_{2}SAT$ : Given a quantified boolean formula $\theta = \exists x_1,...,x_l\forall y_1,...,y_m\psi$, where $\psi$ is a ...
3
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1
answer
161
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complexity class of a function - linear combinations and reductions (Fermionant, immanant, $GL_n$ representations)
The fermionant is a matrix function from physics, which is indexed by a positive integer $k$:
\begin{align}
\operatorname{Ferm}_k(A) = \sum_{\lambda} d_{\lambda}^{(k)} \operatorname{Imm}_{\lambda^T}(A)...
- 31
7
votes
1
answer
226
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What is the complexity class of higher-order primitive recursion?
Short and sweet: The complexity class of primitive recursive functions (WP) is PR (WP, Zoo). What's the complexity class of higher-order primitive recursion (WP)?
The motivating context is simply that ...
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1
vote
1
answer
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Is QMA known to contain Co-NP?
Is QMA known to contain Co-NP? If not, would Co-NP being contained in QMA have any implications for other complexity classes. (e.g. Causing the polynomial heirachy to collapse.)
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1
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Lexicographic Boolean satisfiability
Maximal Satisfying Assignment (Lexicographic Boolean satisfiability / LexMaxSAT), the problem of finding the lexicographically maximum x_1, . . . , x_n ∈ {0, 1}^n that satisfies Boolean formula φ, or ...
3
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0
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112
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Complete problems for $(NP\cap CoNP)/poly$ class and universal representations
It is conjectured that $NP\cap CoNP$ does not have a complete problem with respect to the polynomial-time many-to-one reductions. I would like to know the current knowledge about the nonuniform ...
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0
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195
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What does width $4$ permutation branching program correspond to?
$L$ can be computed by a family of programs over $S_3$
of polynomial length if and only if $L$ can be computed by a family of $MOD3 ◦ MOD2$ circuits of
polynomial size.
$L$ can be computed by a family ...
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2
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0
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206
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Simultaneous evidence for $L\neq NL$ and $P\neq NP$
We believe $L\neq NL$ and $P\neq NP$.
Is there any evidence which simultaneously imply $L\neq NL$ and $P\neq NP$?
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2
votes
1
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63
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Relative error estimation of a special type of GapP function
Consider the functions included in the complexity class GapP.
We know that approximating a function from GapP, in the worst case, to inverse polynomial multiplicative error, is #P-hard. Even correctly ...
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1
vote
1
answer
168
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KRW Conjecture: separation of NC^1 and P
More than a real question this is a recap of something I have been studying. I hope someone will help me getting things straight, so any correction or thought about the following reasoning is more ...
1
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0
answers
209
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anything hinting that EXPTIME $\subseteqq$ PSPACE?
Anything or evidence hinting that $$EXPTIME \subseteqq PSPACE$$？
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0
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72
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Additive error approximations of GapP functions
Consider a GapP function $g(x)$ for $x \in \{0, 1\}^{*}$. Consider an approximation $\tilde g(x)$ such that
\begin{equation}
\left|g(x) - \tilde g(x)\right| \leq \epsilon.
\end{equation}
Consider a ...
- 153
1
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0
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190
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IP, MIP and MIP* with super-polynomial verifier
Regarding each of the above classes, what are the currently known upper bounds when the verifier is given more than polynomial power?
Specifically, when do we reach ALL in each of the above classes ...
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3
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0
answers
124
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Complexity of real coefficients Linear Programs
I would like to know if there are known any polynomial time algorithms for deciding the feasibility of linear programs with real (not integers) coefficients.
I know that for linear programs with ...
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3
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0
answers
131
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Improving the approximation in Stockmeyer's counting theorem
Given a $\#P$ function $f(x)$, we can use Stockmeyer's counting theorem to get an approximation $g(x)$ such that
\begin{equation}
\left(1 - \frac{1}{\text{poly}(n)} \right) f(x) \le g(x) \le \left(...
- 153
2
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0
answers
98
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Consequences of an $o(\log n)$-approximation algorithm for a $\mathsf{\log\text{-}APX}$ hard problem
In [1], Feige proves that if there is a polynomial algorithm with approximation ratio in $o(\log n)$ for any $\mathsf{log\textit{-}APX}$ hard problem (say Minimum Dominating Set), then $\mathsf{NP}\...
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3
votes
1
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What is the best simulation of majority utilizing $\bmod\{2,3,\dots,p\}$ gates?
It is known $AC^0[2]$ cannot get majority function.
Is there a literature on simulation of $MAJ$ function utilizing $AC^0[2,3,\dots,p]$ gates for a finite fixed set of primes $2$ to $p=O(1)$?
What is ...
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0
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73
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On Determinant modulo $2^k$ complexity
Determinant of integer matrix modulo $2$ is complete for the class $\oplus L$. Is determinant modulo $2^k$ computable in $\oplus L$ at any fixed $k$?
How about if $k=o(n)$ where matrix is $n\times n$?
- 12.5k
4
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0
answers
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The graph of problem reductions
A classical approach to study the complexity of a problem $P$ is to efficiently reduce a well known problem $P'$ to $P$, thus showing that $P$ is at least as difficult as $P'$. The TCS literature ...
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1
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156
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Comparing SAT to MCSP reduction class separations and faster SAT class separations?
Assume $SAT$ is in $QuasiP$. We immediately infer $NQuasiP=QuasiP$ and $EXP=NEXP$. From https://people.csail.mit.edu/rrw/easy-witness-nqp.pdf we infer $NQuasiP$ is not in $P/poly$ which implies $...
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0
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54
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Is anything known about NC$^1$ with NP oracle
A few things are known about the class $\textsf{L}$ provided with an $\textsf{NP}$ oracle ($\textsf{L}^\textsf{NP} = \Theta_2^\textsf{P}$ has attracted a bit of attention, for instance [1]) On the ...
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Logarithmic queries to $\Sigma_i^P$ oracle and the Boolean hiearchies
If I understood correctly (the complexity zoo, wikipedia, and some of the cited articles), the class $\textsf{P}^{\textsf{NP}[\log]}$, also known as $\Theta_2^{\textsf{P}}$, sits at the top of the ...
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votes
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321
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Where is $MA$ more relevant than $\exists BPP$?
NP can be defined as the class of languages which admit sets of certificates which are in P. The definition could be as follows.
A language $L$ is in $NP$ iff there is a set $C=\left\{ x,c\right\}$ ...
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$\mathsf{ACC}^0$ and $\mathsf{TC}^0$ with $\mathsf{Cuniform}$-$\oplus\mathsf{L}$ or $\mathsf{Cuniform}$-$\mathsf{NC}^1$ oracle?
$\mathsf{TC}^0$ is a small class with $\oplus\mathsf{L}$ containing it.
Following inclusions are known:
$$\mathsf{Cuniform}\mbox{ -}\mathsf{ACC}^0\subseteq\mathsf{Cuniform}\mbox{ -}\mathsf{TC}^0\...
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2
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1
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Is there any relation between $\mathsf{PPP}\subseteq\mathsf{TFNP}$ and $\mathsf{UP}$? [closed]
A complete problem for $\mathsf{PPP}$ is the pigeon problem which is 'Given a Boolean circuit ${\displaystyle C}$ having the same number ${\displaystyle n}$ of input bits as output bits, find either ...
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1
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Is there a containment between $\mathsf{ZPP}$ and $\mathsf{UP}$?
Do we know if $\mathsf{ZPP}\subseteq\mathsf{UP}$ known or is there oracles against the hypothesis?
- 12.5k
15
votes
2
answers
409
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NLOGTIME versus $\exists$DLOGTIME
$\def\dlt{\mathrm{DLOGTIME}}\def\nlt{\mathrm{NLOGTIME}}\def\mr{\mathrm}$During a recent discussion on another question, I mentioned a factoid $\exists\dlt=\nlt$, but then I realized that I may have ...
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12
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350
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NP complete problem help
I'm currently trying to find a reduction to this problem:
Given a set S of n points (in the plane) in general position, is there a set of at least k triangles (formed using only points in S as ...
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91
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Is it to solid to conclude APX-complete after showing a problem cannot be approximated better than 1.5 and also develop a 2-approximation algorithm
I know the canonical way to show APX-Complete is to give an L-reduction from an already-known APX-complete problem. If I have used gap-preserving reduction to show a problem cannot be approximated ...
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1
answer
152
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What graphs on $\mathbb{N}$ can be encoded as regular languages?
Suppose I represent the natural number 0 by "x", and use the symbol "s" for successor so that I get the following encoding of $\alpha : \mathbb{N} \rightarrow V$ of natural numbers ...
19
votes
1
answer
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Has parameterized complexity led to better algorithms?
I know that for the vertex cover problem, if we know that the parameter $k$ (which is the number of vertices in the solution) is small, then we can expect to solve it feasibly in practice. So far, ...
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3
votes
1
answer
309
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Complexity of type inference in the simply typed lambda calculus
A similar question was answered here:
Is simply typed lambda calculus equivalent to primitive recursive functions
What I conclude from the answers is that the complexity is that of the extended ...
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1
vote
1
answer
128
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Rademacher Complexity of the Composition with an Indicator
Consider the statistical learning setting where you have an arbitrary hypothesis space $\mathcal{H}$, a data space $\mathcal{Z}$, and a bounded loss function $\ell: \mathcal{H}\times \mathcal{Z} \...
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1
vote
1
answer
88
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Can a NEXP machine simulate invalid queries to a promise problem oracle?
Let $A=(A_{YES},A_{NO})$ be some promise problem (such as xSAT, the Local Hamiltonian problem, etc). Suppose we want to show that a P machine with access to a the oracle A can always have its output ...
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1
vote
0
answers
64
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Reference to "compressibility" of logarithmic space [closed]
Is there a reference somewhere for the result SPACE($O(\log n)$) = SPACE($\log n$)? i.e. Big-O doesn't matter in logspace since you can compress the space. I feel like this is an elementary result but ...
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