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Questions tagged [complexity-classes]

Computational complexity classes and their relations

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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
166 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
94 views

When is a problem specified on a TM contained in non-uniform classes such as P/poly? [on hold]

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
215 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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0answers
110 views

Can UNAE3SAT be converted into a P-complete decision problem?

By Self-reducibility, we understand that a search problem can be reduced to the same problem but by a decision problem instead of a function problem. P is trivially self-reducible, but what about P-...
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1answer
79 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 ...
9
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1answer
156 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 ${\bf NP}$...
13
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1answer
423 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
163 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
267 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
68 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
244 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 ...
6
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1answer
168 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
324 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 ...
1
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1answer
243 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
161 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
358 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
403 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 ...
8
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0answers
113 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
181 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}/\...
4
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1answer
287 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 ...
3
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1answer
211 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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0answers
420 views

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 ...
10
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1answer
250 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 ...
4
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1answer
96 views

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 ...
4
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1answer
233 views

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

Is it known that $\textbf{BPP} \subseteq \textbf{P}^{\textbf{NP}}$?
5
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1answer
336 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 ...
8
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1answer
192 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
153 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 ...
1
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0answers
90 views

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 ...
6
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1answer
219 views

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 $\...
2
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1answer
101 views

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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0answers
218 views

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 $BPP^{BQNC}$? In particular is Discrete Logarithm in $BQNC$ or at least in $BPP^{...
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0answers
186 views

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
91 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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0answers
51 views

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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0answers
131 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{...
8
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1answer
92 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 ...
2
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1answer
125 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 ...
3
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0answers
123 views

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 $W[1]$?
5
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1answer
279 views

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
101 views

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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0answers
36 views

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
74 views

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 ...
3
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3answers
499 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
209 views

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 ...
7
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4answers
267 views

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 ...
0
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1answer
345 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
73 views

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
195 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 ...