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Time complexity of decision problems or relations among time-bounded complexity classes. (Use [tag:analysis-of-algorithms] for the time taken by particular algorithms.)

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0answers
36 views

Determine the time complexity [on hold]

I need to determine time complexity of the next code: j1=1 j2=1 for i=1 to n k=j1 while k<=j2 s=s+j k=k+1 j1=j1+2i+1 j2=j2+4i+3 return s ...
8
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0answers
119 views

Time complexity of exponentiating s-sparse matrices

Could someone suggest me a reference which discusses the time complexity of algorithms meant for exponentiating (finding $e^A$ approximately given $A$) s-sparse matrices, along with their error rates? ...
-1
votes
0answers
54 views

Are there any function problems that have worst case superlinear time complexity?

Does there exist a function problem that is proven to have worst case time complexity $\omega(n+m)$ where $n$ is the input size and $m$ is the output size?
7
votes
3answers
159 views

Equivalent formulation of complexity theory in Lambda Calculus?

In complexity theory the definition of time and space complexity both reference a universal Turing machine: resp. the number of steps before halting, and the number of cells on the tape touched. ...
3
votes
2answers
389 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 ...
7
votes
1answer
187 views

TIME(n) versus TIME(nlogn)

The time hierarchy theorem implies TIME($n$) is strictly contained in TIME($n\log^{1+ε}n$) for all ε>0. Is the relationship between TIME($n$) and TIME($nlogn$) known?
3
votes
1answer
95 views

Does a non-constructive proof of bounds of a computable asymptotic complexity, with impossible fix, exist?

Does there exist an algorithm, about which a non-constructive $\omega$-consistent theory $A$ can prove that it has time complexity $O(f(n))$ where $n$ is some univariate function of the input, but ...
2
votes
0answers
89 views

Complexity of solving a polynomial equation

Given a polynomial equation of degree n with m variables, that is guaranteed to have at least one solution, what is would be the ...
5
votes
0answers
99 views

Lower bound for enumerating k closest pair of points

Consider 1 dimensional points $x_1, \dots x_n, x_i \in \mathbb{R}$ where the distance between any two points is defined as $d(x_i, x_j) = \vert x_i - x_j \vert$. The goal is to enumerate all $n^2$ ...
15
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0answers
405 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 ...
3
votes
0answers
40 views

Lower bound for reversing a list using queues

How do you prove (or disprove) that a list of length $n$ cannot be reversed in time $o(n \log n)$ using $O(1)$ queues? Each queue is FIFO. Time refers to the number of operations on the queues. ...
4
votes
0answers
68 views

Finding a largest symmetrical subset of a k-CNF propositional formula

I have a k-CNF propositional formula $F$ which do not admit any global symmetry i.e. there is no permutation $\sigma$ of its variables such that $\sigma(F) = F$ . I'm interested in finding the largest ...
5
votes
2answers
227 views

Data structure to determine if sets are disjoint in o(n) time

My initial question was exactly the title of this post, but after feedback from commenters I have formulated a more precise version of the question that attempts to capture its essence. Does there ...
1
vote
1answer
60 views

Can we replace deterministic part of alternative turing machine with some other equivalent machines?

I'm sorry if it is a low level question but I am so confusing. If $DTime(n)\subseteq \Sigma_2Time(n^{0.2})$ then $DTime(n) \subseteq \Sigma_2DTime(n^{0.2})$ Is this true that $\Sigma_2DTime(n^{0.2})...
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votes
1answer
109 views

What do you think of Bokov's “CNF-SAT is in P” proof? [closed]

There are now several versions of G. V. Bokov's paper "Complexity of the CNF-satisfiability problem", cf. https://arxiv.org/abs/1804.02478 In the most recent version of his paper, the proof is only ...
6
votes
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 ...
2
votes
0answers
47 views

Bit complexity of factoring univariate polynomial over $\mathbb{Q}$ (rationals)

What is the bit complexity of finding all the irreducible factors $f_1, ..., f_r$ of a degree-$d$ polynomial $f(x) = \sum_{i=0}^d a_i\cdot x^i \in \mathbb{Q}[x]$ whose all coefficients are $B$-bit ...
9
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0answers
202 views

How would proof of the Lindelöf hypothesis improve our understanding of computational complexity classes?

A recent press release from the Viterbi School of Engineering at USC discussed the proof of the Lindelöf hypothesis by Athanassios Fokas, a visiting professor from the Department of Applied ...
-1
votes
1answer
49 views

Finding upper and lower bounds of a problem [closed]

We have n balls where 1 is a little heavier than the others and we want to find that heavier ball. We can only put some balls on one side of the scale and some on the other side and see if it leans ...
0
votes
0answers
14 views

Faber polynomials versus other polynomial approximations

Faber polynomials appear to be used to algorithmically approximate the complex exponential function $\exp(-ix)$ for example as in the case of the time evolution in quantum mechanics, for a Hermitian ...
0
votes
0answers
90 views

What does the phrase 'up to logarithmic factors' mean when discussing algorithm time complexity?

I've come across a few papers that discuss the upper bound of algorithmic time complexity and state something to the effect: 'This algorithm solves the problem in $O(\sqrt{N} + \sqrt{M})$, up to ...
6
votes
1answer
333 views

Formalization of proofs and computational complexity paradox?

While reading some articles on formal proofs (see also my previous question on cstheory about the length of ZFC proofs versus human written proofs), I came up with this apparent paradox. Let $M_{...
5
votes
1answer
87 views

Fast Finding Main Diagonal of Matrix Multiplication

Suppose we have two matrices $A_{m\times n}$ and $B_{m\times m}$. Such that $B$ is a symmetric positive definite matrix. Is it possible to compute main diganoal of $A^TBA$ in $O(n\times m)$?
2
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0answers
47 views

Is $SUBLOG\subset DTIME(n)$?

In the course of trying to give a more natural answer to a previous question of mine involving the complexity classes $SUBLOG=\{L\mid L \text{ is recognizable by a sublogarithmic-space TM} \}$ and $...
4
votes
0answers
117 views

Is $L\subset NC^1$

Arora and Barak's online book claims in exercise 6.11 that $NC^1=L$. While the $NC^1\subset L$ direction is relatively straightforward and explained in many other texts, I couldn't prove or find the $...
0
votes
1answer
70 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 ...
7
votes
2answers
281 views

Are space and time hierarchies even comparable?

I am wondering if there are any results to what extent the space and time hierarchies "disagree" on which problem is harder. For example, is it known whether there are languages $L_1$ and $L_2$ such ...
7
votes
2answers
224 views

Reference Request: complexity results on finding $(1+\epsilon) \log n$ size clique in $G(n,1/2)$

I am trying to find results on the best known time complexity for finding $(1+\epsilon) \log n$ sized cliques in $G(n,1/2)$. More general results would be great, i.e. if $C_p$ is the constant such ...
4
votes
0answers
247 views

The problem of whether or not every function computable in time $T(n)$ is computable in time $T(n)^{O(1)}$ and space $T(n)^{o(1)}$ simultaneously

If a function is computable in time $T(n)$, is it computable in time $T(n)^{O(1)}$ and space $T(n)^{o(1)}$ simultaneously? We won't be able to prove it, because it implies the open problems $\text{P} ...
3
votes
1answer
271 views

Most general setting for fine-grained exponential-time complexity classes?

Consider the class of functions computable in time $(b+o(1))^n = 2^{\log_2{(b)} \times n + o(n)}$ on a $2$-tape Turing machine. By the Hennie-Stearns theorem, the same functions are computable in ...
5
votes
1answer
202 views

Examples of quasilinear vs. essentially linear time translatable models

The Hennie-Stearns theorem says that $k$-tape Turing machines with $k \ge 2$ are intertranslatable with loglinear blowup ($O(t \times \log{(t)}$). This would define an equivalence class of models, ...
1
vote
2answers
125 views

What is the time complexity of base conversion on a multi-tape Turing machine?

Base conversion is the problem of converting an integer between representations in two fixed bases. Without loss of generality consider the case of relatively prime bases. I think it's easier to ...
4
votes
0answers
275 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 ...
4
votes
1answer
115 views

What is the time complexity of increasing the precision of finding matrix eigenvalues?

There are various algorithms that output the eigenvalues of an $n \times n$ matrix in time $O(n^3)$. However, I can't find anywhere that tells me about the precision of the output of the algorithm. ...
8
votes
1answer
164 views

Complexity of finding semi-ordered Eulerian tours in a 4-regular graph

I'm trying to figure out the time-complexity of the problem I describe below, which I call the semi-ordered Eulerian tour problem or the SOET problem. Either finding an efficient algorithm for this ...
4
votes
2answers
214 views

When studying the computational complexity of functions $\{0, 1\}^\ast \to \{0, 1\}^\ast$, is it enough to restrict to $\{0, 1\}^\ast \to \{0, 1\}$?

I started reading Avi Wigderson's paper $\mathcal{P}$, $\mathcal{NP}$ and Mathematics – a Computational Complexity Perspective (link). (Notation: $\{0, 1\}^\ast$ is the set of all finite binary ...
3
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0answers
162 views

Complexity of the mandelbrot set on rationals

(Also posted on mathoverflow) Given two rationals $a,b \in \mathbb{Q}$, call $c = a + ib$, i.e., the complex number represented by these two rationals. A point $c$ is contained within the Mandelbrot ...
6
votes
1answer
178 views

Quantum polynomial hierarchy vs counting hierarchy

First of all, I'm kinda surprised that I couldn't find any paper/article defining such hierarchy. It can be defined as follows: $\Delta_0^{\mathsf{BQP}}=\Sigma_0^{\mathsf{BQP}}=\Pi_0^{\mathsf{BQP}}=\...
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votes
1answer
119 views

polylogarithmic space [closed]

I apologize if this is a silly question, but could someone tell me whether the class polyL (polylogarithmic space) is equal to the class ATIME(polylog)? If so, where can I find a reference to this or ...
15
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1answer
272 views

Does $∩_{ε>0} \mathrm{DTIME}(O(n^{2+ε})) = \mathrm{DTIME}(n^{2+o(1)})$?

I expect the answer is no, but I could not actually construct a counterexample. The difference is that in $∩_{ε>0} \mathrm{DTIME}(O(n^{2+ε}))$, we might not be able to pick an $O(n^{2+ε})$ ...
1
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0answers
64 views

Time complexity of polynomial regression with random coefficients

Suppose that I have $$\lim_{k,l,m \rightarrow \infty}(f(x,l) ~~=~~ \sum_{n=1}^{m} g(a_{n,k})h(x,n)$$ where the $a_{n,k}$ are pseudo-random real numbers with $k$ digits generated in an arbitrary way, ...
5
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0answers
337 views

Do we know some quasi-polynomial problem that is known to not be in NP?

The title pretty much says it all, but to explain how I got there: I think, that one of the reasons we are unable to prove or disprove , but mainly disprove $P=NP$ (and yes, I was provoked by the ...
6
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0answers
82 views

Small universal monotone Turing machines

This paper surveys small universal Turing machines. What are some examples of small universal monotone Turing machines, as described by Schmidhuber? Which of these are efficient (polynomial time) ...
0
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1answer
213 views

${\bf NP} \not = {\bf E}$ and ${\bf PSPACE} \not = {\bf E}$

We know that ${\bf NP} \subseteq {\bf PH} \subseteq {\bf PSPACE}$. We also know that ${\bf E} \subset {\bf EXP}$, where ${\bf E} = \cup_c DTIME[2^{cn}]$ and ${\bf EXP} = \cup_c DTIME[2^{n^c}]$. It ...
3
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0answers
119 views

How does communication complexity relate to time complexity in distributed algorithms?

Some distributed algorithms (e.g. Bracha broadcast) runs in a constant number of rounds. I'm interested on how you'd analyse the time complexity of such algorithm, especially when the message size ...
0
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1answer
81 views

Complexity status of restricted k-clique [closed]

Restricted $k$-clique: Input: $(G,v,k)$ where $v$ is vertex in $V$ Output: k-clique containing vertex $v$. What is the space and time complexity status of this Restricted $k$-clique problem? Is ...
11
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1answer
226 views

Comparing two products of lists of integers?

Suppose I have two lists of positive integers of bounded manitude, and I take the product of all elements of each list. What's the best way to determine which product is larger? Of course I can ...
3
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2answers
73 views

Maximal non-reducible vertex cover of a graph

Let $G=(V,E)$ be a graph (i.e. an undirected simple finite graph). We say that a vertex cover $V'$ of $G$ is non-reducible if any $V''$ with $V''\subsetneq V'$ is not a vertex cover of $G$. We say ...
16
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1answer
947 views

“Almost sorting” integers in linear time

I am interested in sorting an array of positive integer values $L = v_1, \ldots, v_n$ in linear time (in the RAM model with uniform cost measure, i.e., integers can have up to logarithmic size but ...
6
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0answers
86 views

Looking for an easy/pedantic exposition of Renegar's famous result on polynomial optimization

In September $1989$, Renegar had this famous sequence of 3 papers titled, "On the Computational Complexity and Geometry of the First-order Theory of the Reals, Part I/II/III". I was wondering if ...