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Questions tagged [complexity-classes]

Computational complexity classes and their relations

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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 ...
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1answer
221 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 ...
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1answer
150 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 ...
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1answer
315 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 ...
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1answer
233 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. ...
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1answer
150 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 ...
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1answer
323 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 ...
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2answers
394 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 ...
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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 $...
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1answer
175 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}/\...
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Proving that a problem is coNP-complete [duplicate]

Let $\le^{p}_{T}$ be the Turing or the Cook reduction and $\le_{m}^{p}$ the Karp-Levin reduction. I know that to prove a problem $P_1$ is coNP-complete I just need to show that $P_1 \in coNP$ and ...
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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 ...
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1answer
190 views

Conesequences of $\forall k\in \mathbb N \space NP\not\subseteq TISP(poly(n),n^k)$

Does $\forall k\in \mathbb N \space NP\not\subseteq TISP(poly(n),n^k)$ has any separation of classes or consequences? My main question is can use this to show that $P \neq NP$ or some thing useful ...
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An algebra of complexity classes

A key feature of unrelativized computation is its composability out of smaller fragments, and to partially capture the composability, I came up with an algebra of fine-grained complexity classes. For ...
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1answer
238 views

Is this game EXPSPACE-complete?

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 a game that will be described below. Let $x$ be ...
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1answer
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Complexity of type-checking in relation to complexity of normalization

In order to verify that a terminating program terminates, one thing that can be done is to actually run the program. That may take a lot of time. If the program is typed in a total type-system, we can ...
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1answer
223 views

Is $\textbf{BPP} \subseteq \textbf{P}^{\textbf{NP}}$?

Is it known that $\textbf{BPP} \subseteq \textbf{P}^{\textbf{NP}}$?
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1answer
319 views

Can one do quantum computing without negative amplitudes?

The typical representation I see of $k$ qubits is a $2^k$ complex numbers $c_i$ for every possible combination of values of those bits, such that the sum of all the squared magnitudes of those numbers ...
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1answer
135 views

How “hard” is it to maximize a polynomial function subject to linear constraints?

General Problem Suppose we have a multivariate polynomial function $f(\mathbf{x})$, and several linear functions $\ell_i(\mathbf{x})$. What is known about the complexity of solving the following ...
4
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1answer
146 views

How powerful is POSIX regex

The set of languages recognized by POSIX regex is a true superset of type 3 languages. But how powerful is POSIX regex really? Is it in an already known class? Is it its own class? If so, what is the ...
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An explicit hard function for P/poly

It is known that $\textbf{MA}_{\textbf{EXP}} \not\subset \textbf{P/poly}$. Is it known any explicit language from $\textbf{MA}_{\textbf{EXP}}$ that does not belongs to $\textbf{P/poly}$? (An example ...
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1answer
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What are the consequences of solving XOR 3-SAT in Logspace?

XOR Formulas Consider boolean formulas with connectives $\wedge$ (AND) and $\oplus$ (XOR). Such a boolean formula is a valid instance for XOR SAT if it is a conjunction of $\oplus$-clauses. An $\...
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1answer
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What is the best approximation and exact algorithm for vertex cover on cubic graphs?

"Best" = best performing in terms of run-time for exact algorithm and approximation ratio for an approximation algorithm.
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BQNC and Abelian Hidden Subgroup Problem

We know integer factorization is in $BPP^{BQNC}$ from Cleve and Watrous. Is Abelian Hidden Subgroup Problem also in $BPP^{BQNC}$? If not what is it that makes integer factorization special? In ...
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Primality in $NC$ hierarchy?

AKS primality testing solves whether a given integer is prime in $P$. AKS algorithm is following: Input: integer n > 1. Check if $n$ is a perfect power: if $n = a^b$ for integers $a > 1$ and $b &...
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1answer
74 views

Is Prime Bounded Quadratic Congruence NP-complete?

Bounded Quadratic Congruence: Instance: Three positive integers $a$, $b$ and $c$. Question: Is there a positive integer $x<c$ such that $x^{2} \equiv a \, (mod \ b)$? Bounded Quadratic ...
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An alternative characterization of some NExp-Time Turing machine with oracles

Let me denote by $\Sigma_i^P$ be a class from i-th level of polynomial time hierarchy (see eg. PH). I'm interested in the following type of a Turing Machine $\mathcal{M}$: $\mathcal{M}$ is ...
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108 views

Is there any NC-complete problem with respect to logspace reduction?

The question is on the title. We all know that $\text{NL}$ and $\text{P}$ have such problems. So I wonder the same thing about $\text{NC}$. More interestingly, is there any $k \ge 2$ and any $\text{...
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1answer
87 views

$BPL$ with polylog random bits is in $L$

Consider a $BPL$ machine (namely, a probabilistic algorithm that uses logspace and polynomially many random bits). It is known (Saks-Zhou) that $BPL \subseteq DSPACE(log^{1.5}(n))$. My question is ...
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1answer
121 views

Best $\Pi_k \text{SAT}$ running time?

Define $\Pi_k \text{SAT}$ by 'Given a quantified boolean formula $\varphi = \forall y_1\exists y_2\dots Q_ky_k\mbox{ }\phi(y_1, \dots, y_k)$, where $\phi(y_1, \dots, y_k)$ is boolean predicate with ...
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What are the consequences if $W[i]=W[i-1]$?

$FPT=W[1]$ does not collapse the $W$ hierarchy however falsifies $ETH$ belief. Is there non-trivial consequence if $W[i]=W[i-1]$ and any other consequence at $W[1]$?
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1answer
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Are there languages in $DTIME(2^{t(n)})$ that are not in $NTIME(t(n))$?

Are there languages in $DTIME(2^{t(n)})$ that are not in $NTIME(t(n))$? This question came up while watching "Graduate Complexity at CMU - Lecture 2: Hierarchy Theorems (Time, Space, and ...
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0answers
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Proof of Sipser-Lautmann Theorem

I have written the following answer as an attempt to prove a variation of Sispser-Lautmann theorem, but it was rejected without any comments. I would appreciate if anyone can find the flaws in this ...
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Monotone complexity of PLP

Blum and Nisan show Positive Linear Programming could be done in $NC$ if we only ask for approximate solutions. This paper https://pdfs.semanticscholar.org/8dc7/5aa9d72864022d848c3e599c5f24d9d527e7....
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1answer
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Constrained Topological Sorting with bounded number of chains

In general, constrained topological sorting is NP-hard. Now we add another constraint to it, such that take any k+1 nodes and there will be at least one pair ...
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3answers
462 views

Why is the circuit class AC0 unavoidable?

Take AC0. What is a natural thought process that leads to the definition of AC0? Does this class arise intrinsically anywhere? My problem is that in the case of unbounded fan-in, AND and OR gates ...
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2answers
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Efficiently computable by a “simple” algorithm?

I am interested in the relation between "program complexity" and "computational complexity". In particular, I was wondering What is known about the minimal length a program must have to solve a ...
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4answers
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What are semantic classes that have a syntactic equivalent?

This question is related to Benefits for syntactic and semantic classes. As mentioned there, $\mathsf{PSPACE} = \mathsf{IP}$, which can be interpreted as the semantic class $\mathsf{IP}$ obtaining a ...
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1answer
315 views

What are the problems in EXPSPACE \ EXPTIME?

Based on the diagram below (from Papadimitriou's book) we can see that PSPACE ⊆ EXPTIME ⊆ EXPSPACE. We know that PSPACE ⊊ EXPSPACE, which means that there are problems that can be solved in polynomial ...
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1answer
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Reduction from SAT to binary matrix subset problem

I'm trying to understand the answer by @Denis on this question, where he showed a way to reduce from SAT to the binary matrix column subset selection problem. The reason I'm having to post a new ...
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2answers
186 views

If only pathological cases of NP-hard problems are difficult to solve, then why isn't NP-hard defined to only include those pathological cases?

NP-hard problems are not used in cryptography, because they are believed to be computationally-intractable in the worst case but are not computationally-intractable in the average case. Is there a ...
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Find a pair of nodes with maximum sum of distances in k given trees

For k edge-weighted trees $T_1,T_2...T_k$ which contain the same set of nodes $\{1,2,... n \}$, I want to find a pair of nodes $(x,y)$ which maxifies $$\sum_{i=1}^k d_i(x,y)$$ where $d_i(x,y)$ ...
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1answer
449 views

Complexity of testing if two sets of $m$ points in $\mathbb{R}^n$ differ only by rotation?

Imagine we have two size $m$ sets of points $X,Y\subset \mathbb{R}^n$. What is (time) complexity of testing if they differ only by rotation?: there exists rotation matrix $OO^T=O^TO=I$ such that $X=OY$...
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1answer
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PromiseBQP and expectation values of operators

This question is regarding The Equivalence of Searching and Sampling by Aaronson. In page 4 he makes the following statement, ... a difficult and unsolved meta-question is whether PromiseBPP = ...
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2answers
708 views

Isn't it trivial to represent any classical Physics problem in a Spin-Glass format which is NP-Complete?

In the late 80's there were several efforts to use Spin-Glass models to formulate other computational problems such as: Protein Folding and Neural Networks. Wouldn't it be straight forward to reduce ...
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Kolmogorov generic oracle

In Does the Polynomial Hierarchy Collapse if Onto Functions are Invertible?, authors defined a new type of generic oracles named Kolmogorov generic oracles. They proved following results relative to $...
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1answer
367 views

Complexity of modal logic IK5

What is the complexity of local satisfiability problem for modal logic $\mathit{IK5}$? Herein we denote by $IK5$ the modal logic over euclidean frames extended with inverse modality. Could you provide ...
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64 views

Using idea of entropy (maybe Shannon entropy or other continuous entropy) to the topic of functional analysis

I am an electrical engineer without a detailed background in theoretical computer science. I am posting here since I hypothesize that the concept of entropy or other branches of information theory (as ...
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0answers
87 views

Does the NP-hardness of finding any valid solution imply NPO-hardness?

Suppose we have an NP optimization problem $A$ such that the problem of finding any valid solution, regardless of whether it is good or bad, is NP-hard (as it happens in NPO-complete problems such as ...
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Should GCT focus on $PSPACE\not\subseteq P/poly$?

GCT tries to show $P$ is not $NP$ by showing $NP$ is not in $P/poly$. Could it be useful in showing $\Sigma_{i+1}\not\subseteq P^{\Sigma_i}/Poly$ at every $i>0$? Suppose if it turns out that $\...