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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.

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My friend and I were trying to reduce the 3-SAT problem to the Sudoku problem for a university project. To do that, we first attempted to reduce 3-SAT to triangle edge decomposition. However, we ran ...
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I am thinking about a 2-player game that some authors call Node Kayles. It's played on a graph and the players build a maximal independent set together: They alternate choosing vertices to add to an ...
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I’ve been exploring a measurement approach for NP and NP-complete problems based on average time per logical step. I define: ...
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I'm interested in the following problem: given a (multi-)graph with each edge coloured by one of 3 colours, find a perfect matching with exactly k_i edges of colour i in {1,2,3}. I'm also interested ...
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I want to know a lower bound for the complexity of the decision problem for $\langle \mathbb{Z}; + \rangle$. The below paper notes that Presburger arithmetic, originally $\langle \mathbb{N}; +\rangle$,...
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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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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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Let $\langle W_e : e\in\mathbb{N}\rangle$ be the standard effective enumeration of recursively enumerable (r.e.) sets, where $$ n\in W_e \;\Longleftrightarrow\; \exists s\;\big(\varphi_e(n)\ \text{...
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Goal Prove that $f(n) = a_pn^p + a_{p-1}n^{p-1} + ... + a_1n + a_0$ is $\Theta(n^p)$ Issue I am having trouble proving $f(n)$ is $\Omega(n^p)$. I know I need a $c_0$ and $k$ such that $f(n) \ge c_0n^p$...
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I am interested in solving the general Cauchy problem: $$\begin{cases}\frac{dx}{dt}=f(x, t) \\ x(t_0)=x_0\end{cases}$$ computationally. Of course, I know there are plenty of well-established methods ...
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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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In case of a finite subset of a group, the subgroup test boils down to showing that the subset is closed under the group operation. This holds, in particular, for the subsets of a finite group. Q. ...
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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$. I was reading Expander Graphs and their Applications (Lecture notes for a course by Nati Linial and ...
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A necessary condition for a subset $\Sigma\subseteq S_n$ to be a transitive permutation group of order $n$, is to be... transitive. Is the best algorithm to check $\Sigma$'s transitivity faster than ...
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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 ...
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Hi I am trying to improve my error function . I have some data that is in form of nested tuple. These tuple nesting is base on importance of the data (all the lowest depth data is in as real numbers). ...
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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 ...
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I was reading a paper on a certain reducibilities in Polynomial classes. And I stumbled across this definition: For sets $A,B$ computable in polynomial time, we write $A \leq^{\#} B$ via a polynomial $...
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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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In studying algorithms, I encountered the following exercise: Show that $\sum_{i = 1}^n O(i)$ is not $O(n)$. I understand that to solve it one just has to realize that $\sum_{i = 1}^n i = \frac{n(n+...
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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 $$ ...
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Just out of curiosity, I have created a sorting algorithm that is a mix between bogosort and selection sort. It operates as follows: Initialise $x$ to 1. This will count the number of sorted elements....
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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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People who know the semantics of Łukasiewicz logic may skip to the ‘the question proper’ part of the description. A reminder on the semantics of Ł Variables are evaluated over $[0,1]$; there are ...
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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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I was thinking about the following question: Alice and Bob play a game. There is a set $X$ of $p^k$ elements known to both of them, $p$ being a prime number. Alice has two binary operations on the set,...
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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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I was learning about one-way functions in cryptography and it occurred that it might make sense to consider a category of computable functions. Is this a thing? For example, we could define a category ...
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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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I wanted to verify whether my counter-example for proving that CFLs are not closed under Perfect shuffle is indeed correct. I just want to know if the counter-example is correct, I know the proof can ...
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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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Both of the problems in the title have a decision version which is NP-hard. My question is whether if SVP can be computed for a tractable example, can a minimum distance codeword for a related linear ...
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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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I'm looking for a theorem like the invariance theorem but for a non-universal case: finite functions. Suppose I have a language L describing finite functions $f : M \to N$ for all $M, N \in \mathbb{N}$...
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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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I am planning to self-study Computational Complexity Theory. After some research, I have narrowed down the following two books: Rudich/Wigderson—Computational Complexity Theory Arora/Barak—...
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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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Problem: Given a string $S$ of length $n$, find the longest palindromic substring. If there are multiple such substrings of the same maximum length, return the lexicographically smallest one. My ...
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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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Consider a triangular grid with n rows, where the i-th row contains i points forming a triangular array — the shape is analogous to that of an n -row Pascal triangle. Each point is initially ...
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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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The analysis for the average-case of Insertion-sort, is given here, with C-implementation. It states the chance for the $i-$th insertion, requiring $0, 1, 2,\cdots, i-1$ comparisons is equal, and is ...
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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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As a network engineer working on optimization problems, I've observed that the literature suggests convex separable programming problems are not significantly more difficult to solve than linear ...
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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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This is a problem in Computational Complexity: A Modern Approach exercise 6.7, but I don't have any idea. Here are some informations may be useful: I know that P=NP implies that P=NP=PH the hint in ...
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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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Let $T = \mathbb{Z}^N$, i.e. $T$ is the universe of $N$-dimensional integer vectors, each vector being denoted as $\vec{x} = \begin{bmatrix} x_1 & x_2 & \ldots & x_N \end{bmatrix}$. Let $...
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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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Suppose we have an integer $n$ and two lists $L_1,L_2$ of equal length $k$ containing integers. In my setting $k$ is subexponential in $\log(n)$, so $k$ is quite a bit smaller than $n$, but not too ...
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