Questions tagged [complexity-classes]
Computational complexity classes and their relations
549
questions
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38 views
Randomized Reduction for Maximization Problem
I have two maximization problems $P_1$ and $P_2$ where the decision version $L_1 = \{(x, t) : \operatorname{Val}_1(x)\ge t\}$ of $P_1$ is $\mathsf{NP}$-complete. Let $f:P_1\to P_2$ be a randomized ...
-3
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0answers
59 views
Violating the hierarchy theorems mildly
We know $NC\subsetneq PSPACE$ unconditionally.
Do we know $PSPACE\subseteq NC/log$ is possible?
Do we know $PSPACE\subsetneq NC/log$ is possible?
Likewise take two classes $A\subseteq P$ and $B\...
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0answers
45 views
Parallel Tractability
The class $\textsf{NC}$ consists of problems that can be solved on a PRAM in ${(\log{n})}^c$ rounds (parallel time), where $n$ is the size of the problem instance.
Is this considered to be the "...
2
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0answers
83 views
What is the difference between ${\bf P ^{\bf FewP}}$ and ${\bf UP ^{\bf FewP}}$?
What is the difference between ${\bf P ^{\bf FewP}}$ and ${\bf UP ^{\bf FewP}}$ ? In other words, what is the language that can be decided by ${\bf UP ^{\bf FewP}}$, but cannot be decided by ${\bf P ^{...
4
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0answers
74 views
BPP fragment of a PSPACE complete problem
Consider a PSPACE-complete problem (e.g., TQBF).
Is there a sub-problem in BPP, that is not known to be in P?
Is there a general technique of finding such sub-problems? Are any of them "natural" (i.e....
5
votes
1answer
174 views
If NP in BPP then NP equals RP
I am looking for a reference to the fact that if NP is included in BPP then NP is equal to RP. See for instance these links:
https://cs.stackexchange.com/q/80509
http://www.inf.ed.ac.uk/teaching/...
1
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1answer
71 views
The decision procedure of theory of closed real field is in NP-hard?
The decision procedure of theory of closed real field refers to https://en.wikipedia.org/wiki/Decidability_of_first-order_theories_of_the_real_numbers
4
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122 views
Terminology: FNP, with P replaced by NP?
Consider these two classes of search problems:
Search problems with poly-sized solutions s.t. verifying solutions is in P.
Search problems with poly-sized solutions s.t. verifying solutions is in NP.
...
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67 views
Does Descriptive Complexity techniques have the naturalisation barrier?
I wished to know if the proof attempts at separation of complexity classes via the methods outlined by Descriptive Complexity theorists naturalise?
By naturalise I'm talking about the Idea of Natural ...
3
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1answer
94 views
Term for a set that is not immune
At the outer bounds of computational complexity classes are those defined through computability theory (AKA recursion theory). This is where we get the well known complexity classes such as R, RE, and ...
4
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1answer
295 views
Which complexity class does this problem belong to?
Consider the following problem $\mathcal{P}$.
Instance: A Boolean formula $F$ of $n$ Boolean variables ($x_1,...,x_n$) and $m$ Boolean parameters ($b_1,...,b_m$) where $0 \leq m \leq n$.
Problem: ...
3
votes
1answer
137 views
Co-NP definition
The class Co-NP is defined as all the languages $L$ such that $\overline{L} \in NP$. An example that appears in the book of Arora and Barak is of $\overline{SAT}$, which is defined as $\overline{SAT} =...
1
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1answer
69 views
Are there common names for the subtiers of PTIME?
We all know P, or PTIME, I think, as a common name for the class of polynomial-time problems. Are there common names for the first few levels inside P; that is, for constant-time, linear-time, ...
4
votes
1answer
157 views
On complexity class $\mathsf{\Pi_2 L}$
I suggest the following definition of $\mathsf{\Pi_2 L}$ (similarly to the certificate definition of $\mathsf{NL}$):
A language $L$ belongs to $\mathsf{\Pi_2 L}$ iff there exists a deterministic ...
10
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0answers
179 views
On Courcelle's question about Monadic second-order logic with cardinality predicates
I have found the following question at openproblemgarden.org:
The problem concerns the extension of Monadic Second Order Logic (over a binary relation representing the edge relation) with the ...
8
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0answers
143 views
A canonical complete problem for EXP and NEXP in terms of formulae
3SAT is a complete problem for NP. TQBF is a complete problem for PSPACE.
Is there direct way to define canonical complete problems for EXP and NEXP in terms of boolean formulae? I have only seen ...
10
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0answers
296 views
Is unary $\Pi_2$-SUBSETSUM coNP-complete?
Consider the following problem:
for given integers $a_1, \ldots, a_{2n}$ and $A$ that are given in unary representation
define is it true that
for every $S \subseteq \{1, ..., 2n \}$ such that $|...
19
votes
2answers
623 views
PPAD and Quantum
Today in New York and all over the world Christos Papadimitriou's birthday is celebrated. This is a good opportunity to ask about the relations between Christos' complexity class PPAD (and his other ...
5
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1answer
133 views
Consequences/existence of problems without any “optimal” algorithm
Let $P$ be some kind of "problem" such as addition or graph coloring, that has an input size $n$. Let $S_P$ denote the set of algorithms $A_1, A_2, \dots$ which deterministically solve $P$. Based off ...
10
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1answer
624 views
A game on several graphs
Consider the following game on a directed weighted graph $G$ with a chip at some node.
All nodes of $G$ are marked by A or B.
There are two players Alice and Bob. The goal of Alice (Bob) is to ...
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1answer
77 views
what does NP ⊆ DTIME(…) mean?
Recently I've seen inside theory of a paper. This time complexity, DTIME, is completely new for me. Can somebody explain it?
Also, the paper shows that the misinformation containment problem cannot ...
6
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1answer
104 views
Separation of AM and SZK
Are any results on the separation of AM from SZK known (e.g. relativized separation, or a separation assuming one-way functions exist, etc.)?
2
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1answer
157 views
Possibility of hierarchy with $UP$ class?
I am not sure if this is a cheap query. However I am unable to find this myself. So I am posting here. The standard complexity class is built with $NP$ and $coNP$ and leads up to $PSPACE$. The ...
0
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1answer
163 views
Language in $PSPACE$ and not necessarily in $P$ if $P=PP$?
If $P=PP$ then the counting hierarchy collapses to $CH=P$. Because so many complexity classes are contained in $CH$, this causes most classes to now be contained in $P$. My question is whether this is ...
2
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0answers
60 views
$XP_{\text{uniform}}=FPT$ and update to $EPTAS$ section in complexity zoo?
Complexity zoo in https://complexityzoo.uwaterloo.ca/Complexity_Zoo:E#eptas has the following:
$FPT = XPuniform\implies EPTAS = PTAS$.
Fundamentals of Parametrized complexity on page $534$ has
...
2
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0answers
139 views
Is the following problem in $coNP$?
Given an $n\times n$ matrix $M$ with $\mathbb Z$ entries is 'does an $\frac n2\times\frac n2$ minor of $M$ vanish?' in $\bf{coNP}$?
At least one $\frac n2\times\frac n2$ minor non-vanish implies rank ...
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1answer
87 views
Why is $BPP^{NP}$ in polynomial hierarchy? [closed]
Why is $BPP^{NP}$ in the polynomial hierarchy?
I know that $BPP$ is contained in $NP^{NP}$, so $BPP$ is inside $PH$. However, how does that imply $BPP^{NP}$ is inside the polynomial hierarchy?
7
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1answer
249 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)?
3
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1answer
188 views
String theory based computations
I was reading Arora and Barak's book on computational complexity and in the section on 'criticism on Turing machine model and the class P' along with quantum computer it also mentions possibilities of ...
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0answers
120 views
Canonical complete problem for $\mathrm{FP}^{\Sigma^p_2}$
Given a $\Sigma^p_2$-complete oracle (i.e., $\Sigma_2 \mathrm{SAT}$), I have a problem that requires to call this oracle polynomially many times and returns an integer. Essentially, this is a function ...
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114 views
Can we define a meaningful concept of exptime reductions (as opposed to polytime reductions) for classes like NEXP or NEEXP?
Typically we are only interested in polytime reductions as we are usually interested in showing a reduction from one NP-problem to another.
However, if we consider larger complexity classes such as ...
11
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0answers
198 views
Computational Complexity of the Frobenius Problem
The Frobenius problem takes as input $n$ positive integers $a_1,\ldots,a_n$ with $\gcd(a_1,\ldots,a_n)=1$ and asks for the largest integer $F$ that cannot be written in the form $F=a_1x_1+a_2x_2+\...
1
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2answers
106 views
When is a problem specified on a TM contained in non-uniform classes such as P/poly? [closed]
In this paper by Gottesman and Irani: https://arxiv.org/abs/0905.2419 , they prove NEXP-hardness of tiling an $N\times N$ grid. They do so by encoding a TM in the tiles making up the grid. However, ...
6
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1answer
241 views
Example problem that is not in $2^{o(n)}$ but could be solved in $O(2^{cn})$ for any $c > 0$ (suggested by wording of ETH)
In the wikipedia article on Time Complexity it is written that:
The exponential time hypothesis (ETH) is that 3SAT, the satisfiability problem of Boolean formulas in conjunctive normal form with, ...
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1answer
88 views
Are there any known languages in the intersection of NP and co-NP but not in P? [closed]
We currently don't know the relationship between NP and co-NP, but would it be possible to show whether the intersection is equal to P? I can't think of any languages in both NP and co-NP, but not in ...
10
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1answer
169 views
Natural candidates for NP-E and E-NP
It has been known since the early 70's that ${\bf NP}$ and ${\bf E}=DTIME(2^{O(n)})$ are not equal (because ${\bf E}$ is not closed under polynomial-time many-one reductions, in contrast to ${\bf NP}$...
14
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0answers
449 views
Is there a P-complete language X such that succinct-X is in P?
I came across a paper called "A Note on Succinct Representation of Graphs". It seems that in the discussion section they claim that for any problem $X$ that is $\mathrm{P}$-hard under projections, $\...
0
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3answers
249 views
Hamiltonian cycle vs co-NP [closed]
I am trying to understand co-NP and its implications properly.
The French Wikipedia page describing co-NP provides the "complementary" version of the Hamiltonian cycle in co-NP as follows:
...
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0answers
277 views
EXPSPACE proof and its implications
I'm dealing with the min-max regret 0-1 Integer Linear Programming problem (MMR-ILP, for short), which is formulated as below.
\begin{equation} \label{eq:nip_obj}
\min_{x \in \Phi} \sum_{i = 1}^n ...
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0answers
70 views
Complexity of enumerating over promise problems and circuits?
Given an enumeration over all Turing Machine which run with increasing length, is there a ``complexity class'' which describes the complexity of determining whether a given TM satisfies the promise ...
11
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1answer
263 views
Is DSPACE(n) = DSPACE(1.5n)?
From space-hierarchy theorem it is known that if $f$ is space-constructible then
DSPACE($2f(n)$) is not equal to DSPACE($f(n))$.
Here, by DSPACE($f(n))$ I mean the class of all problems that can ...
7
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1answer
191 views
Does Max Planar 3-SAT admit a PTAS?
Suppose we are given a formula $\phi$ of 3-SAT, with variables $x_1,\dots, x_n$ and clauses $C_1,\dots, C_m$. Consider the graph $G_\phi$ where there is one node for each clause $C_i$, for each ...
7
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1answer
334 views
“Berman-Hartmanis Conjecture Separates NP From All Super-Poly. DTIME Classes” — Worthy of arXiv.org?
Do you believe this paper is worthy of arXiv.org? I have searched via Google, and to my knowledge, no one else has this result. I'm not asking you to fully scrutinize the paper, I'm just asking if you ...
2
votes
1answer
269 views
$P=BPP$ without good PRGs?
We know that the existence of good pseudorandom generators (PRGs) does not only imply $P=BPP$, but also $PromiseP=PromiseBPP$.
Let us assume $PromiseP\ne PromiseBPP$. Then good PRGs do not exist. ...
8
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1answer
188 views
Type theory and computational complexity
Is there a type system, which restricts the lambda terms to the terms which fall inside a complexity class? Like the typable terms in the theory are strictly inside the complexity class ? Or is it not ...
10
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1answer
370 views
What is the complexity of this game?
This is a generalization of my previous question.
Let $M$ be a polynomial-time deterministic machine that can ask questions to some oracle $A$. Initially $A$ is empty but this is can be changed after ...
3
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2answers
414 views
Is there a non-deterministic version of the complexity class PP?
From a quick skim of the literature (and complexity zoo), there doesn't seem to be a non-deterministic version of PP. Is there a reason for this (e.g. PP=non-deterministic PP?)
Edit: Perhaps I ...
9
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0answers
130 views
Complexity of fractional SAT
Let $(a, k)$-SAT be $k$-SAT with the promise that if there is there is a satisfying assignment, then there is such an assignment that satisfies at least $a$ literals of every clause. Can 3-SAT with $...
5
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1answer
192 views
Boolean circuits which correspond to L/poly
Branching programs are usually used as a computation model for non-uniform logarithmic space $\mathsf{L}/\mathrm{poly}$.
Is there a reference about Boolean circuits corresponding to $\mathsf{L}/\...
6
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1answer
417 views
Indications that strengthen the conjecture: NEXP ⊊ EXP^NP
I am trying to find indications that strengthen the conjecture of NEXP ⊊ EXP^NP.
Clearly NEXP ⊆ EXP^NP, and there are some hints that this inclusion is proper.
Some Examples:
1. A paper by Shuichi ...
