Questions tagged [computational-complexity]
Use for questions about the efficiency of a specific algorithm (the amount of resources, such as running time or memory, that it requires) or for questions about the efficiency of any algorithm solving a given problem.
3,578 questions
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Does the Interior Point Method Solve Klee-Minty Cubes adversarial instances in Polynomial Time?
I am a network engineer currently studying optimization problems. Out of curiosity, I was fascinated by the fact that the Simplex Method has an exponential worst-case complexity, a property famously ...
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Proof that NP is a subset of EXP
I would like to clarify a misunderstanding I have about the proof that all NP problems can be solved in exponential time. The argument as I understand it is that you can simply test all possible ...
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Prove Big Theta for degree p polynomial
Goal
Prove that $f(n) = a_pn^p + a_{p-1}n^{p-1} + ... + a_1n + a_0$ is $\Theta(n^p)$
Issue
I am having trouble proving $f(n)$ is $\Omega(n^p)$. I know I need a $c_0$ and $k$ such that $f(n) \ge c_0n^p$...
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How does this algorithm "rank" amongst other in the literature?
By merging together the contributions from: a) this answer, b) the comments under this answer, we come up to the following:
Claim. For $n\in\mathbb N$, let $Q=(\{1,\dots,n\},*)$ be a quasigroup. Then, ...
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Prove that the zig-zag product of $G$ and $H$ (where $H$ is the smaller of the two) lifts $H^2$
Prove that the zig-zag product of $G$ and $H$ (where $H$ is the smaller of the two) lifts $H^2$.
I was reading Expander Graphs and their Applications (Lecture notes for a course by Nati Linial and ...
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Polynomial-Time Algorithms for Canonical Form of Ternary Matrices under Row/Column Permutations and Column Negations
We study equivalence classes of ternary matrices of size $m\times n$, where equivalence is defined via row permutations, column permutations, and negation of entire columns. Our goal is to define and ...
- 1,860
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Enumerating all "elementary" cycles on a graph
I am interested in enumerating all possible "elementary" cycles of a given graph $G=(V,E)$. What I mean by elementary here, is a notion that I have but am not sure what its called in ...
- 141
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Cost bounds for computing GCD of multivariate polynomials
Where to find the cost bound formulae for computing GCD of multivariate polynomials?
I see such for $\mathbb{Z}[x]$ in "Modern Computer Algebra" by Gathen & Gerhard.
But I need the case ...
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VC-Dimension for finite domain and maximum $k$ hypothesis class
In "Understanding-Machine-Learning" by Shalev-Shwartz and Ben-David Section 6.8 Exercise 2.2 one has to determine $\text{VCdim}(\mathcal{H})$, where the hypothesis class is defined as
$$
...
- 147
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Expressing $1$ of a finite field
Assume we have a finite field $F$ with $p^k$ elements. We start with a single non-zero element $x \in F$, and in one operation we can get the sum or product of any two elements we already have. We ...
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Evaluating $\Pi_i\Pi_j\Pi_k(1 + a_ib_jc_k)$ efficiently
I'd like to efficiently evaluate $\Pi_i^N\Pi_j^N\Pi_k^N(1 + a_ib_jc_k)$ without enumerating the $N^3$ terms by brute force.
I was able to find an approach that achieves $O(Nlog^2(N))$ for the 2-...
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Fast algorithm to compute restricted double sum
I stumbled across the following question: Can one compute the double sum
$$
\sum_{i = 1}^n \sum_{j:\ j <i}\ 1\{a_j > a_i\} 1\{b_j > b_i\}
$$
in $O(n \log(n))$, where
$$
(a_i)_{i = 1,..,n}, \...
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Can the CVP → OptCVP reduction be extended to lattices with real (possibly irrational) basis?
I'm studying the reduction from the Closest Vector Problem (CVP) to its optimization variant (OptCVP) as presented in Theorem 8 of these lecture notes by Prof. Micciancio.
The original reduction ...
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approximation algorithm for MAX-CLIQUE
Suppose we have given a 0.12 approximation algorithm for MAX-CLIQUE is an efficient algorithm that on an input graph G with optimal solution of size 𝑘, returns a clique of size at least 0.12⋅𝑘.
My ...
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is there a Polynomial‑time reduction from k‑Dominating Set in arbitrary graphs to k‑Dominating Set in regular graphs?
I’m studying the complexity of the Dominating Set problem under degree constraints. It’s well known that deciding whether a graph
G has a dominating set of size
k is NP‑complete in general.
If such a ...
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70
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What function grows slower than any exponential but faster than "most" sub-exponentials?
I'm looking for a function $f \colon \mathbb{N} \to \mathbb{N}$ (or $\mathbb{R}^+ \to \mathbb{R}^+$) that satisfies:
Sub-exponential growth: For every $a > c1$ for some positive c1, $f(n) = o(a^n)$...
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Example of a string with many longest palindromic substrings under a specific inequality constraint
Let $s$ be a string of length $n$, and let $L$ be a fixed integer. Suppose the following conditions hold:
The string $s$ contains exactly $k$ palindromic substrings of length $L$ (they may overlap),
...
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Is there a relationship between the Kolmogorov complexity of an invertible function and its inverse?
Given a function $R$ that can be described with a minimal length binary program, its Kolmogorov complexity is the length of that program.
If the function is invertible, can we make some statements ...
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Number of comparisons, in the best-case for the Binary-search based insertion-sort.
This post is further to the post here.
The analysis for the number of comparisons made by binary-search, for the average-case of Insertion-sort, is stated below. Also, C-implementation code is stated ...
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Checking if a space of binary matrices has a solution generating an isotropic subspace
In general, checking whether a system of bilinear equations over $\mathbb{F}_2$ has a solution is NP-hard, but I was wondering if the special structure of this problem allowed us to do better: Let $A$ ...
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How can we derive the formula for the minimal strict decision nodes in array Minimum-Finding?
I am searching for a formula that determines the lowest number of strict decision nodes for finding one or multiple minimums from a random array of pre-determined size.
The only input of the formula ...
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Reducing the number of special symbols in mapping reduction of Modded PCP to post's correspondence problem
Background: there exists a proof for a mapping reduction of modified PCP(MPCP) to PCP (in MPCP you always start the answer sequence with the first domino). This proof (can be found online, from ...
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Finding order of time complexity for program finding the $m$-subsets of an $n$-set.
For the code here, the analysis for the order-of-time complexity is as follows:
For the purpose of finding the time-complexity of the above program; the program statements of concern are:
...
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Actual Meaning of Big $O$, $\Theta$, and $\Omega$ (Computer Science)
I learned from my university the following about big $O$, big $\Theta$, and big $\Omega$ notations and I would like to verify if what I learned is correct or not because all the other definitions I ...
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LP relaxation leads to exponential Branch & Bound
Consider the following LP program
$$ \min x_{n+1} $$
subject to:
$$ \sum_{i=0}^n 2 x_i + x_{n+1} = n $$
$$ x_i \in \{ 0, 1 \} $$
And $n$ odd.
The claim is that using the standard B&B algorithm ...
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