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Questions tagged [time-complexity]

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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Is there a relation between BBH (black box hypothesis) and SETH (strong exponential time hypothesis)?

Is there a relation between BBH (black box hypothesis) and SETH (strong exponential time hypothesis)?
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46 views

Complexity of comparing extended integer power towers

Inspired by this stackexchange question, is it an open problem to compare two power towers of positive integers if we additionally allow numbers lower in the tower to themselves be represented by ...
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73 views

Finding 3SUM witness when promised a solution

Suppose we have a 3SUM instance given with the promise that there exists at least one solution. Is the trivial $O(n^2)$ (modulo logarithmic improvements) solution still the best algorithm or is there ...
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Arranging sets in a hierarchy

Suppose you have sets $S_1, \dots S_m$ such that $\sum_i |S_i| = n$. The goal is to arrange all the sets into a (possible unconnected) DAG such that $S_i$ is a parent (or ancestor) of $S_j$ iff $S_j \...
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how is time complexity defined in computational learning theory

In general, when we say an algorithm $A$ PAC learns $C$ in time $t$, we say $A$ takes time $t$ before outputting a hypothesis $h$, and the hypothesis can be evaluated (on every $x$) in time $t$. Now ...
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Testing emptiness property complexity in Sum of Squares Proof systems

Take the set $$\mathcal T=\{f_1(x_1,\dots,x_n)=\dots=f_m(x_1,\dots,x_n)=0, h_1(x_1,\dots,x_n)\geq a_1,\dots,h_t(x_1,\dots,x_n)\geq a_t\}$$ where $$h_1(x_1,\dots,x_n),\dots,h_t(x_1,\dots,x_n)\in\mathbb ...
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34 views

Time complexity of finding a point of infinite order on a rank 1 elliptic curve over Q

As an outsider, it sounds like a lot of progress has been made on understanding rank 1 elliptic curves over Q. Much of the BSD conjecture is known for rank 1, and Heegner points provide a way in ...
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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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Parallel building time of a k-d tree on n points with n processors

Given a point set with $n$ points to build a k-d tree on. We have $n$ processors available. What is the time-optimal building time for the k-d tree? A straight forward parallelization would be as ...
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34 views

Minimize The Number of Connected Components in Hit-map of A Boolean Matrix

Suppose there is a matrix with the value of 0 and 1. The hit-map of the matrix (0 is blue and 1 is red) create some connected component (see the following figure as an instance): Is there any ...
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Worst case polynomial in elimination theory under rank conditions?

Given $n$ polynomials $h_1(x_1,\dots,x_{2n}),\dots,h_{2n}(x_1,\dots,x_{2n})\in\mathbb Z[x_1,\dots,x_{2n}]$ where each of $h_1(x_1,\dots,x_{2n}),\dots,h_{2n}(x_1,\dots,x_{2n})$ is homogeneous of degree ...
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Evaluating addition chains

I hope this is a suitable place to ask this question. An addition chain of size $n$ is a sequence $x_1, \dots, x_n$, where $x_1$ is fixed to 1 and $x_i = x_j + x_k$ for some $j,k < i$. I am ...
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269 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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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? ...
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3answers
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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. ...
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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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1answer
209 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?
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1answer
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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 ...
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116 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 ...
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0answers
106 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 $n^2$ ...
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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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46 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. ...
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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 ...
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2answers
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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 ...
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1answer
70 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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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 ...
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1answer
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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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51 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 ...
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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 ...
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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 ...
7
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1answer
338 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_{...
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1answer
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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 $O(n\times m)$?
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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 $...
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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 $...
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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
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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 ...
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2answers
232 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 ...
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0answers
251 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} ...
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1answer
273 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 ...
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1answer
220 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, ...
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2answers
133 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 ...
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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 ...
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1answer
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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. ...
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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 ...
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2answers
216 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 ...
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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 ...
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1answer
197 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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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 ...
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1answer
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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 $O(n^{2+ε})$ ...
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72 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, ...