Questions tagged [time-complexity]

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

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35 views

Fastest known algorithm to enumerate k-cliques in a graph for fixed k

Is the best known algorithm for finding all $k$-cliques in a graph with $n$ nodes, for a fixed $k$, given by https://theory.stanford.edu/~virgi/combclique-ipl-g.pdf ? The time-complexity of the ...
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Runtime trial division [closed]

I am studying primality testing. It is claimed that trial division takes O(n) operations. For large primes it is therefore infeasible. It is claimed that trial division is not polynomial. But is not O(...
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On the classes $E$, $DTIME$ and $NTIME$?

For every $c>0$ is there a $c'$ such that $DTIME(n^c)^E\subseteq DTIME(2^{n^{c'}})$ hold and vice versa for every $d'>0$ is there a $d$ such that $DTIME(2^{n^{d'}})\subseteq DTIME(n^d)^E$ hold? ...
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Complexity of two promise graph coloring problems

I. Consider the problem 'Given a chromatic number $k$ graph on $n$ vertices then for all coloring with $k$ colors is there always a color with at least $m$ vertices?'. What is the known complexity ...
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Rigorous evidence for difficulty of sub-half exponential lower bound

Razborov provided rigorous evidence for non-natural proof necessity for superhalf-exponential lower bound for Discrete logarithm problem. Is there any rigorous evidence for difficulty of super ...
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Complexity of computing Earth Mover's Distance when the costs satisfy the triangle inequality

Let p and q by two categorical probability distributions over $\{1,2,...,k\}$. Given a set of costs $c_{ij} \ge 0, i,j \in \{1,2,...,k\}$ that satisfy the triangle inequality, that is $c_{ij} \le c_{...
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39 views

Does Kruskal's algorithm Have an Every-case Complexity O(ElogE)? [closed]

I ask this as a learning exercise as I'm about to graduate (just wrapped up my last final : ] ) and the topic is interesting to me. Not to mention I have seen posts around where people who have been ...
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Is 4-in-a-row PSPACE-complete?

This paper by Laurens Kuiper shows that axis-parallel k-in-a-row is PSPACE-complete in complexity for k ≥ 5, but leaves the question open for k = 4. Has there been any research progress on this ...
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2answers
141 views

What are those deterministic algorithms for k-SAT that are not derandomization of random algorithms like PPSZ and Schöning's local search?

I am doing a survey on k-SAT where time complexity is in terms of n, i.e. the number of variables in a formula. As for the fast algorithms for k-SAT, we see biased-PPSZ, PPSZ, Schöning's local search,...
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1answer
163 views

Time complexity for multiplying two lower triangular matrices

I was wondering, if multiplication of two $n \times n$ lower (or upper) triangular matrices has a more efficient algorithm than multiplication of two general $n \times n$ matrices? $$ \begin{bmatrix} ...
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Program size versus program running time

Short "naive" question: Is it true that faster algorithms require longer programs ? Given a decision problem $A$ and a reasonable model of computation, there can be many ways (algorithms) to ...
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1answer
66 views

Running an algorithm for fixed amount of time on RAM model machine

Suppose there is a deterministic algorithm of size $O(1)$ that operates on an input of size $N$ on a RAM model machine. I want to run the algorithm for $O(\sqrt{N})$ time, pause the algorithm, print "...
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Is PP invariant under changing its cut-off from 1/2 to another number?

Suppose I have a fixed family of quantum circuits $\{C_i\}$ for which determining whether the maximum output acceptance probabilities are $p\geq 1/2$ or $p< 1/2$ is PP-hard. Now suppose I have the ...
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Are there complexity theory consequences of the collapse NEXP=EXP^NP?

It is clear that $NEXP\subseteq EXP^{NP}$, as a TM with exponential run time can simply query the NP oracle with an exponentially long query. However, it's not clear that the reverse $EXP^{NP}\...
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1answer
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Complexity of solving systems of linear equations with hash preimages

Introduction: I'm researching a decision problem that I thought was in NP because there are certificates for its instances that have a polynomial number of elements. However, I realized that there are ...
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Sorting using comparisons that are not simple mappings of simple comparisons

The Python language has a sort(x) function that sorts a list based on the intrinsic comparison operator associated with the type of the elements of its input list x. One can also provide a cmp ...
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1answer
157 views

Are there problems in $DTIME(n^k) - DTIME(n^{k-1})$ that are not hard for $DTIME(n^{k-1})$ under nearly linear time reductions?

Background It can be challenging to find computational problems that are solvable in $DTIME(n^k) - DTIME(n^{k-1})$ where $k \geq 2$. Although some natural problems are known to exist, many of them ...
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2answers
166 views

Problem in deterministic time $n^p$ and not lower

I'm looking for any language $L$ candiate to be in $DTIME(n^p) -DTIME(n^{p-1})$ (it takes at least $n^{p-1}$ steps to determine if an input is in L with a 2-tape $TM$, but L is polynomially solvable). ...
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1answer
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Determining if a word of specific length exists that is not accepted by a NFA

It is known that the problem of determining if an NFA accepts every word is PSPACE-COMPLETE, meaning it is also NP-Hard, but is this weaker version of the problem still NP-hard? Given an NFA and a ...
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185 views

Fast algorithms for evaluating functions with high Kolmogorov complexity

Motivation: I am motivated by a concrete example that occurs in neuroscience, dendritic computation, which may be approximated by functions computable on binary trees [1]. To be more precise, I ...
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Complexity of multi-objective optimization problems

How can we define and prove the worst-case complexity of multi-objective optimization problems (MOOP)? It is easy to see that, if one of the objectives is an NP-Hard optimization problem, then the ...
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1answer
122 views

Complexity of unbalanced bipartite isomorphism

For $i=1,2$, let $G_i=(A_i\cup B_i,E_i)$ be an undirected bipartite graph with bipartition $A_i$ and $B_i$, where $|A_1|=|A_2|=a$ and $|B_1|=|B_2|=b$ with $a\le b$. Question. Is the problem of ...
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1answer
101 views

Induction on all polynomial runtimes?

Has there ever been a proof technique to show that a language isn't in $\mathrm{P}$, by showing inductively there isn't any $k$ for which the language is in $\mathrm{TIME}(n^k)$? e.g.: $L\notin \...
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What is the run-time of LP?

Are there any further generalizations known to the result about run-time of a LP than what is stated in Theorem 1 of these lecture notes, https://nisheethvishnoi.files.wordpress.com/2018/05/lecture71....
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1answer
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Are there common names for the subtiers of PTIME?

We all know P, or PTIME, I think, as a common name for the class of polynomial-time problems. Are there common names for the first few levels inside P; that is, for constant-time, linear-time, ...
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A sandwich Algorithm / Data Structure [closed]

$O(n^c)$ is asymptotically greater than $O(\log^d n) $ for all possible pair of values of $c$ and $d$. Can you give an example of a problem (or data structure) which has running time (or query/...
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1answer
157 views

Evidence integer multiplication is in linear time?

After millenia of quest we have identified two $n$ bit integers can be multiplied in $O(n\log n)$ time. Please refer details in https://www.quantamagazine.org/mathematicians-discover-the-perfect-way-...
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58 views

When can convex optimization be considered to be exactly solvable?

If one is trying to find the global minima of a convex function using gradient descent then one will get a run-time which is a function of $\epsilon >0$ where $\epsilon$ measures the accuracy of ...
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164 views

Complexity of extracting a coefficient of a polynomial in multiple variables

I'm looking for efficient algorithms for problems of the following type: Let's say we have the variables $x_1,...,x_n$. Over these variables, we are given a function $p_1\cdot ... \cdot p_m$, ...
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What is a natural problem in theory of computation?

In Stephen Cook's paper on the P vs NP problem,[1] he states the following [2]: Feasibility Thesis: A natural problem has a feasible algorithm iff it has a polynomial-time algorithm. My question ...
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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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1answer
134 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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191 views

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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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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274 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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69 views

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
146 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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2answers
328 views

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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1answer
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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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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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34 views

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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3answers
323 views

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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1answer
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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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263 views

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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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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0answers
84 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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43 views

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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78 views

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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40 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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1answer
245 views

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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