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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54 views

How to prove a general convex set is nonempty or empty in polynomial time?

The general convex set should be represented by a set of (generalized) inequalities $f_{i}(x)\leq 0 $ with $ f_{i}(x) $ being convex in $ x $. I know ellipsoid method and interior method, but I do ...
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87 views

Consequences/existence of problems without any “optimal” algorithm

Let $P$ be some kind of "problem" such as addition or graph coloring, that has an input size $n$. Let $S_P$ denote the set of algorithms $A_1, A_2, \dots$ which deterministically solve $P$. Based off ...
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Nondeterminstic Linear Time vs Other Complexity Classes

Is it known whether or not nondeterministic linear time contains $P$ and/or smaller classes such as Uniform-$NC^1$?
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Is BQP upper bounded by the class of problems computable by an exponential number of GPUs?

Consider the class $EXPGPU$ that informally contains all problems that can be stocastically solved in polynomial time by an exponential number of processors. Question: Is $BQP \subset EXPGPU$? My ...
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what does NP ⊆ DTIME(…) mean?

Recently I've seen inside theory of a paper. This time complexity, DTIME, is completely new for me. Can somebody explain it? Also, the paper shows that the misinformation containment problem cannot ...
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Is counting simple cycles in $P$ for graphs of bounded tree width?

Motivation: Determining if a graph has a Hamiltonian cycle is $NP$-hard in general. However, determining if there is a Hamiltonian cycle is in polynomial time on graphs of bounded tree width, either ...
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Hidden Constants in Complexity of Algorithms

For many problems, the algorithm with the best asymptotic complexity has a very large constant factor that is hidden by big O notation. This occurs in matrix multiplication, integer multiplication (...
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184 views

Does Depth-First-Search admit a quasilinear time algorithm in mutitape Turing Machine model?

Depth-First-Search (DFS) has a quasilinear (i.e.,$\widetilde{O}(m+n)$) time algorithm in random access model (RAM). I am curious about whether DFS still admits a $\widetilde{O}(m+n)$ time algorithm in ...
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Cost of in-place partitioning integer arrays

Suppose we are given an array $a\colon[n]\to[m]$ of length $n$ (and each entry is between 1 and m). We will denote the $i$th entry of the array as $a[i]$. Task: Permute the array $a$ in-place so that ...
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1answer
123 views

Given a subset of of the hypercube and an affine transform of it, find the affine map

This is a follow up to this resolved question. Suppose we are given a set of bitvectors $A\subseteq\mathbb{F}_2^d$ and an invertible affine transformed copy of it $$B=\{Mx + s\mid x\in A\}$$ for some ...
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Given a subset of the hypercube and a copy translated by s, find s

Problem: Suppose we are given an $n$ element subset $A\subseteq\{0,1\}^d$ of the $d$ dimensional hypercube and a translated copy $B= A+s$ by some secret $s\in\{0,1\}^d$. Find $s$ as fast as possible ...
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Faster than linear set intersection

Is there a probabilistic data structure, such as the bloom filter, for which we could perform intersection superlinearly (at the cost of false positives)? (I only know of a O(n) time complexity set ...
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“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 ...
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Techniques to improve the efficiency of Dynamic Time Warping Algorithm

I am analyzing a set of time series that are shifted along the x-axis (see image below for clarification). I intend to average the time series and for that I would like to overlap all the start points ...
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Why is $BPP^{NP}$ in polynomial hierarchy? [closed]

Why is $BPP^{NP}$ in the polynomial hierarchy? I know that $BPP$ is contained in $NP^{NP}$, so $BPP$ is inside $PH$. However, how does that imply $BPP^{NP}$ is inside the polynomial hierarchy?
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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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2-NEXPTIME-complete problems

We have a problem and we found an algorithm that appear to be 2-nexptime. I would like to find known 2-nexptime-complete problems in order to find a lower bound. I found in literature mainly two ...
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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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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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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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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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Counting the number of vertex covers: when is it hard?

Consider the #P-complete problem of counting the number of vertex covers of a given graph $G = (V, E)$. I'd like to know if there is any result showing how the hardness of such problem varies 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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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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What notable automaton models have polynomially-decidable containment?

I'm trying to solve a particular problem, and I thought I might be able to solve it using automata theory. I'm wondering, what models of automata have containment decidable in polynomial time? i.e. if ...
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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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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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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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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
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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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1answer
213 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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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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108 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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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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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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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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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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One Stack, Two Queues

background Several years ago, when I was an undergraduate, we were given a homework on amortized analysis. I was unable to solve one of the problems. I had asked it in comp.theory, but no ...
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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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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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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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Complexity of eigenvalue problem

Many matrix diagonalization algorithms have time complexity $\mathcal{O}(n^3)$ where $n$ is the number of columns/raws (consider only square matrices). What is the best time lower bound it is known? ...
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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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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 ...