Questions tagged [complexity-classes]
Computational complexity classes and their relations
604
questions
0
votes
1answer
160 views
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 ...
6
votes
1answer
187 views
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 $...
6
votes
1answer
255 views
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 ...
0
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0answers
73 views
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 ...
7
votes
1answer
174 views
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
votes
1answer
156 views
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)...
7
votes
1answer
191 views
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 ...
1
vote
1answer
135 views
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.)
0
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1answer
164 views
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
votes
0answers
104 views
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 ...
4
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0answers
189 views
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 ...
2
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0answers
196 views
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 $P\neq NP$?
2
votes
1answer
55 views
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 ...
1
vote
1answer
152 views
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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0answers
203 views
anything hinting that EXPTIME $\subseteqq$ PSPACE?
Anything or evidence hinting that $$EXPTIME \subseteqq PSPACE$$？
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0answers
67 views
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 ...
1
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0answers
183 views
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 ...
3
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0answers
122 views
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 ...
3
votes
0answers
109 views
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(...
2
votes
0answers
96 views
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}\...
3
votes
1answer
127 views
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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0answers
73 views
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 $n\times n$?
4
votes
0answers
158 views
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 ...
0
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1answer
142 views
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 $...
0
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0answers
74 views
What is the best reduction we know from flavors of $SAT$ to $MCSP$?
Consider $\mathcal P(n,m)$ to be a class from the set $\{k\mbox{-}\mathsf{SAT}(n,m),\mathsf{CIRCUITSAT}(n,m)\}$ where $k$ is fixed, $n$ is number of variables and $m$ is number of clauses.
Denote $\...
0
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0answers
53 views
Is anything known about NC$^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 ...
2
votes
0answers
53 views
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 ...
5
votes
1answer
225 views
Where is $MA$ more relevant than $\exists BPP$?
(EDITED)
A previous version of this question asked about a complexity class I called $MA^*$, which has been recognized by users to be $\exists BPP$.
The difference between $MA$ and $\exists BPP$ is ...
0
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0answers
112 views
$\mathsf{ACC}^0$ 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\...
2
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1answer
117 views
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 ...
5
votes
1answer
249 views
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?
15
votes
2answers
394 views
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 ...
12
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0answers
344 views
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 ...
3
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0answers
75 views
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 ...
2
votes
1answer
147 views
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
1answer
1k views
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, ...
2
votes
1answer
184 views
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 ...
1
vote
1answer
95 views
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} \...
1
vote
1answer
86 views
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 ...
1
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0answers
64 views
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 ...
1
vote
1answer
95 views
Consequences of turning $\oplus \text{SAT}$ into few satisfying assignments
Suppose there is a reduction which, given a $\oplus \text{SAT}$ instance $\phi$, returns another $\oplus \text{SAT}$ instance $\psi$ having all the following properties:
The size of $\psi$ is ...
6
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0answers
101 views
Counting on grid graphs
Are there problems defined on graphs, such as counting 2-factors, Hamiltonian cycles, connected spanning subgraphs etc., that are in $\#P$ and remain hard for grid graphs?
Since there seem to be ...
6
votes
1answer
142 views
Complexity of approximating a real function using queries
Consider the following computational problem, where $I$ is the real interval $[-1,1]$:
There is a monotonically-increasing function $f: I\to I$. You are allowed to access it only through queries of ...
-5
votes
1answer
140 views
Is mathematical proof itself NP-hard?
At the 8:00 mark of this video, he claims that proving things is itself an NP problem. I'm looking for more insight into this. Could someone help explain this concept to me and also provide a link to ...
3
votes
1answer
142 views
Are all RegExp solvable in O(n)?
I'm wondering if all features, that are often part of modern RegEx engines, are solvable in O(n). I'm talking about features like repeating patterns ([abc]+);\1 ...
1
vote
1answer
134 views
Diagonalization arguments for QMA type proof systems
Diagonalization is a very common technique to find oracle separations. For example, it can be used to separate $\cal{P}$ and $\cal{NP}$, with the essential idea being that of constructing an oracle in ...
4
votes
0answers
146 views
Why can MIP be restricted to just two provers?
In several places I see it referred to that the MIP class can be assumed to be two interactive provers that don't communicate with each other, rather than any polynomial number of provers. Why are ...
12
votes
2answers
297 views
Is there an assumption that implies $P=ZPP$ which is not known to imply $P=BPP$?
There are assumptions that are known to imply that $P = BPP$. For example, if there exists a function in $E = DTIME(2^{O(n)})$ that has circuit complexity $2^{\Omega(n)}$, then $P = BPP$ [1]. Clearly, ...
7
votes
1answer
188 views
Complexity class of efficient streaming algorithms
Consider the class of problems $\mathsf{StreamL}$ which can be solved in logarithmic space reading the input in a single pass from left to right. In other words:
$L \in \mathsf{StreamL}$ if there ...
5
votes
0answers
124 views
Is there an analogue of QMA where Merlin gives Arthur unitaries rather than states?
$\def\braket#1#2{\langle#1|#2\rangle}\def\bra#1{\langle#1|}\def\ket#1{|#1\rangle}$Is there an analogue to $\mathsf{QMA}$ where Merlin provides to Arthur single-use access to a unitary operator $U$? By ...
