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Questions tagged [complexity-classes]

Computational complexity classes and their relations

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How could the No Free Lunch Theorem be applied to complexity classes

I was reading a chapter from a book, which states: (section 11.3.4) No Free Lunch (NFL) has not been proven to hold over the set of problems in the complexity class NP. I was confused because NP ...
12
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1answer
344 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, $\...
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1answer
358 views

Implications between $\mathsf{L}=\mathsf{P}$ and $\mathsf{NL}=\mathsf{NP}$?

If we can prove that $\mathsf{L}=\mathsf{P}$, does it imply that $\mathsf{NL}=\mathsf{NP}$ ? I thought it is the case, but I cannot prove it (also for the converse).
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Prove that language in L

How to prove that $L = \{(\phi,x) | \space Boolean \space formula \space \phi \space is \space true \space on \space a \space set \space of \space x\} $ lies in L?
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Prove that if A is NP-complete and B is coNP-complete, than AxB is NP-, coNP-complete

AxB means cartesian product of A and B. May someone help me with this? I even have no idea how to prove that AxB belongs to NP or coNP
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3answers
153 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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2answers
563 views

Validity implies NP=#P? [closed]

Valid progams for NP imply every solution is a valid answer. NP not equals #P implies not all solutions are answers. Therefore, Validity implies NP=#P. NP is the problem class for ...
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0answers
177 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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1answer
187 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 ...
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0answers
205 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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1answer
286 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 ...
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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
235 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
88 views
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1answer
159 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
235 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
321 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
338 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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1answer
154 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 ...
13
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1answer
349 views

Problems in NC not known to lie in NC2

Are there interesting problems that are in $\mathsf{NC}$ but not known to be in $\mathsf{NC^{2}}$? In the paper 'A Taxonomy of Problems With Fast Parallel Algorithms', Cook mentions that MIS was known ...
3
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2answers
397 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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0answers
112 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 $...
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1answer
177 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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1answer
322 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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0answers
61 views

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 ...
5
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1answer
276 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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11answers
114k views

Is Norbert Blum's 2017 proof that $P \ne NP$ correct?

Norbert Blum recently posted a 38-page proof that $P \ne NP$. Is it correct? Also on topic: where else (on the internet) is its correctness being discussed? Note: the focus of this question text has ...
3
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1answer
203 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
416 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 ...
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1answer
227 views
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1answer
244 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
312 views

Ruzzo-Simon-Tompa oracle access mechanism

In a paper on relativizing logspace computations, Ladner and Lynch construct an oracle relative to which $\mathsf{NL} \nsubseteq \mathsf{P}$. There are some more pathological examples in this vein in ...
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1answer
82 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 ...
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1answer
326 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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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
493 views

Problems in BQP but conjectured to be outside P

Wikipedia listed four problems that are in $BQP$ but conjectured to be outside $P$: Integer factorization; Discrete logarithm; Simulation of quantum systems; Computing the Jones polynomial at certain ...
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2answers
710 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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1answer
147 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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0answers
89 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 ...
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1answer
208 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 $\...
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4answers
1k views

Examples where the uniqueness of the solution makes it easier to find

The complexity class $\mathsf{UP}$ consists of those $\mathsf{NP}$-problems that can be decided by a polynomial time nondeterministic Turing machine which has at most one accepting computational path. ...
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0answers
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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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1answer
122 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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1answer
84 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
114 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
90 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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0answers
120 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 $W[1]$?
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0answers
100 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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1answer
73 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 ...
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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....