Questions tagged [complexity-classes]
Computational complexity classes and their relations
596
questions
10
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1answer
338 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 ...
5
votes
1answer
209 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
votes
1answer
250 views
Is $\textbf{BPP} \subseteq \textbf{P}^{\textbf{NP}}$?
Is it known that $\textbf{BPP} \subseteq \textbf{P}^{\textbf{NP}}$?
5
votes
1answer
464 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 ...
9
votes
1answer
353 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 ...
6
votes
1answer
186 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
vote
0answers
99 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
votes
1answer
376 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 $\...
3
votes
2answers
207 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.
2
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0answers
1k 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^{...
3
votes
0answers
232 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 &...
0
votes
1answer
121 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 ...
2
votes
0answers
262 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{...
9
votes
1answer
125 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
votes
1answer
135 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
votes
0answers
140 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
votes
1answer
374 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 ...
1
vote
0answers
121 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 ...
1
vote
0answers
37 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....
1
vote
1answer
82 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 ...
4
votes
3answers
914 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 ...
1
vote
2answers
232 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
votes
4answers
295 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
votes
1answer
758 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 ...
0
votes
1answer
88 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 ...
2
votes
2answers
244 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 ...
4
votes
0answers
200 views
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 $d_i(x,y)$ ...
12
votes
1answer
590 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 $X=OY$...
1
vote
1answer
97 views
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 = ...
8
votes
2answers
1k views
Isn't it “trivial” to represent/reduce any classical physics problem into a Spin-Glass which is NP-Complete?
In the late 80's there were several highly cited efforts to use Spin-Glass models to formulate other computational problems such as: Protein Folding and Neural Networks.
Isn't it straight forward to ...
1
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0answers
64 views
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 $...
8
votes
1answer
432 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 ...
1
vote
0answers
103 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 ...
-5
votes
1answer
173 views
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 $\...
10
votes
1answer
199 views
On sparse complete sets and P vs L
Mahaney's Theorem tells us that if there is a sparse $NP$-complete set under polynomial-time many-one reductions, then $P = NP$. (See "Sparse complete sets for NP: Solution of a conjecture of Berman ...
3
votes
0answers
75 views
How to charactorize computational complexity based on finding solution to algebraic equations? [closed]
The Matiyasevich/MRDP Theorem relates two notions:one from computability theory, the other from number theory, thus Turing Machine and algorithms finding integral solution to algebraic equations can ...
1
vote
1answer
260 views
Relation between transcendental numbers and computational complexity?
Regarding the relation, there is the Hartmanis-Stearns conjecture, but beyond Turing Machine of the realtime output, there is no further conjecture or theorem. Obviously irrational algebraic number ...
7
votes
1answer
153 views
Presburger arithmetic: is it known to be in $EXPSPACE \setminus EXP$?
My understanding of the Presburger arithmetic decision problem is that it requires doubly-exponential time, but only singly-exponential space, meaning that it is in $EXPSPACE \setminus EXP$. ...
4
votes
0answers
159 views
Random self reducibility and NP
I was reading the Wikipedia page Random self-reducibility and it states:
If an NP-complete problem is non-adaptively random self-reducible the polynomial hierarchy collapses to $\Sigma_3$.
I am ...
2
votes
0answers
64 views
On reduction between two classes?
https://link.springer.com/article/10.1007/s00153-013-0351-x gives seven reductions $m,c,d,p,btt(1),\ell,tt$.
What does norm $1$ mean in $btt(1)$?
Is there illustrative examples that help understand ...
2
votes
1answer
127 views
What is conjunctive truth table reduction?
What are conjunctive/disjunctive truth table reductions and how do they compare with other reductions?
10
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0answers
115 views
Problems in $\mathsf{NC^{2}}$ that are not known to be in $\mathsf{AC^{1}}$ or $\mathsf{DET}$
Do we know of any problems in $\mathsf{NC^{2}}$ that are not known to be in $\mathsf{AC^{1}}$ or $\mathsf{DET}$?
Context: based on Josh's answer to this question, it could be possible that all ...
1
vote
1answer
100 views
Recursively presenting or even enumerating all P-hard languages
A class of languages $C$ is recursively presentable if there is an effective enumeration of Turing machines $\mathcal{M}_1,\mathcal{M}_2,\ldots$ such that $C=\{L(\mathcal{M}_i)\mid i=1,2,\ldots\}$. ...
4
votes
2answers
265 views
On $NP$, $\oplus P$ and $PP$?
We know $\oplus P^{\oplus P}=\oplus P$, $PP^{\oplus P}\subseteq P^{PP}$ and $NP\subseteq PP$.
Is $\oplus P^{PP}=PP$?
Why is it difficult to show $NP^{NP}\subseteq PP$?
What is the smallest known ...
2
votes
0answers
36 views
A class of languages admitted by a class of grammars equivalent to $\mathbf{PR}$?
Is there a class of languages $L(G)$ admitted by a class of phrase structure grammars $G$ equivalent to $\mathbf{PR}$? (the class of primitive recursive languages = $\mathbf{LOOP}$)?
In greater ...
-1
votes
1answer
135 views
Two queries related to Toda
Is the Isolation lemma crucial for $PH\subseteq BPP^{\oplus P}$ theorem and would avoiding the Isolation lemma say anything more that is not known?
3
votes
0answers
128 views
Approximation class of finding decision trees with minimal depth
We are given some sets $S_1, \cdots , S_n$ and two disjoint sets $A$ and $B$. A decision tree is a binary tree where each node asks "$x \in S_i? $" for some $i$, taking the left branch means "yes", ...
-4
votes
1answer
63 views
Proving NP-complete problem
Suppose the following problem:
Given an undirected graph G=(V,E), is it possible to choose a subset V' of vertex set V, such that deleting it removes all triangles (cycles of length 3), where |V'| is ...
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0answers
76 views
On collapsing the Exponential time hierarchy
Define $\Sigma^E_0 = \Pi^E_0=E$,
for every $n>0$, define $\Sigma^E_n=NE^{\Sigma^p_{n-1}}$,
for every $n>0$, define $\Pi^E_n=CoNE^{\Sigma^p_{n-1}}$.
Define the Exponential time hierarchy by $EH=\...
5
votes
0answers
555 views
What happens when PSPACE contains NEXP?
The complexity class NEXP is defined as the set of all languages that an arbitrary nondeterministic exponential time Turing Machine accepts (i.e. NTIME($2^{p(n)}$) for $p()$ a polynomial).
In the ...
