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.)
312
questions
2
votes
0answers
34 views
3SUM Complexity—A special(?) Case
In the paper “Consequences of Faster Alignment of Sequences” by
Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann which appeared in ICALP 2014 and is available here the following version of ...
7
votes
2answers
320 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 ...
-2
votes
0answers
47 views
Probabilistic exponential time [closed]
Is it somewhat explored the exponential time complexity of bounded error probabilistic algorithms?
Is there a name for the exponential analogue of BPP?
In particular I wonder if it can be proved that ...
-4
votes
0answers
33 views
How to determine if given “complex” time complexity is O(n^2)?
If a time complexity, for example or or , is not determinable by just looking at it whether is it in class O(n^2) or not, how do I decide? If a time complexity is given, and in it there are more ...
3
votes
0answers
102 views
Graph problems in P with unknown lower bounds
I am looking for references to interesting graph problems, which are known to be in P, but their precise big-O lower bounds are elusive. I would split this into 2 classes:
problems, where we know of ...
8
votes
1answer
300 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 ...
3
votes
0answers
61 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 ...
2
votes
0answers
56 views
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_{...
3
votes
2answers
147 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,...
2
votes
0answers
328 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 ...
9
votes
0answers
116 views
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 ...
0
votes
1answer
193 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?
$$
\begin{bmatrix}
...
5
votes
0answers
122 views
Program size versus program running time
Short "naive" question:
Is it true that faster algorithms require longer programs ?
Given a decision problem $A$ and a reasonable model of computation, there can be many ways (algorithms) ...
2
votes
0answers
95 views
Is $SUBLOG\subset DTIME(n)$?
In the course of trying to give a more natural answer to a previous question of mine involving the complexity classes $$SUBLOG=\{L\mid L \text{ is recognizable by a sublogarithmic-space TM} \}$$ and $...
1
vote
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 "...
2
votes
0answers
96 views
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 ...
3
votes
1answer
494 views
Time complexity of d-dimensional convex hull
Consider the convex hull problem in $\Re^d$:
Input: a list of $n$ points $S$ in $\Re^d$,
Output: the vertices of the convex hull of $S$.
What is the best lower bound on the time complexity of ...
0
votes
0answers
82 views
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}\...
2
votes
1answer
112 views
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 ...
7
votes
1answer
153 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$ ...
2
votes
1answer
160 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 ...
2
votes
2answers
168 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).
...
1
vote
0answers
18 views
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 ...
15
votes
2answers
4k views
What is the fastest algorithm to compute rank of a rectangular matrix?
Given an $m \times n$ matrix (assuming $m \ge n$), what is the fastest algorithm to compute its rank and basis of the columns?
I am aware it can be solved through linear matroid intersection, which ...
6
votes
1answer
92 views
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 ...
2
votes
0answers
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 ...
0
votes
0answers
60 views
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 ...
3
votes
1answer
126 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 ...
1
vote
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 \...
0
votes
0answers
60 views
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....
2
votes
1answer
75 views
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, ...
-5
votes
2answers
189 views
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/...
-2
votes
1answer
160 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-...
0
votes
0answers
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 ...
18
votes
11answers
2k views
Are there any problems whose best known algorithm has run time $O\left(\frac{f(n)}{\log n}\right)$
I've never seen an algorithm with a log in the denominator before, and I'm wondering if there are any actually useful algorithms with this form?
I understand lots of things that might cause a log ...
5
votes
0answers
166 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$, ...
11
votes
2answers
3k views
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 ...
1
vote
0answers
99 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 ...
5
votes
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 ...
6
votes
0answers
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$?
-4
votes
1answer
115 views
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 ...
4
votes
3answers
329 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 ...
9
votes
1answer
411 views
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 (...
5
votes
0answers
278 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 ...
2
votes
0answers
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 ...
4
votes
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 ...
7
votes
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 ...
16
votes
2answers
2k 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 ...
1
vote
0answers
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 ...
-1
votes
1answer
94 views
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?
