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This is a branch that includes: computational complexity theory; complexity classes, NP-completeness and other completeness concepts; oracle analogues of complexity classes; complexity-theoretic computational models; regular languages; context-free languages; Kolmogorov Complexity and so on.

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I have a fixed $n \times n$ matrix $M$ whose entries are all either 0 or 1. I want to compute the product $Mv$ for various vectors $v \in \mathbb{R}^n$ (or over other fields/rings). Since $M$ is fixed ...
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I recently read in the paper "Quantum NP is hard for PH" by S. Fenner et. al. that "graph non-isomorphism is known to be in coC_=P", but they did not attach a reference. I have ...
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This is related to the second part of this old question of mine. For $A\subseteq\mathbb{N}$ let $\mathfrak{N}_A=(\mathbb{N};0,1,+,A)$ be the expansion of Presburger arithmetic with (a predicate naming)...
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Given a network $G=(V,E,w)$, two vertices $s$ and $t$ as source and sink, and a designated path $P_0$ from $s$ to $t$, the inverse shortest path problem (ISP) asks to adjust new weights $w^*$ at ...
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The idea described below came to my mind and I couldn't find anything similar out there (although I didn't try too hard to be honest). So my questions are: "What is it" and "How is this ...
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One of the strongest results on the decidability of theories is Rabin's Tree Theorem. One way to state it is the following: tThe problem of deciding whether a sentence on the monadic second order (MSO)...
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I'm working on a project that requires quickly calculating the resultant of two polynomials in $\mathbb{Z}[x]$ of large degree $d,e$. On the Wikipedia article for polynomial resultants, it says that ...
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Say that a $k$-ring is a ring in which $x^k=x$ for all $x$, and write $m\trianglelefteq n$ iff every $m$-ring is an $n$-ring. It's not hard to show (see the end of this answer) that $\trianglelefteq$ ...
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I have a multiprojective variety $X$ in a product of projective spaces given by a multigraded ideal $I$. Suppose that the multiprojective variety is embedded into a product of projective spaces the ...
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Let the following data be given. Two positive integers $m$ and $n$. A family of sets $B_a \subseteq \{1, \dots, n\}$ (for $a \in \{1, \dots, m\}$). The task is to count the number $N$ of injective ...
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Consider the following decision problem, which we will call COMPARE. We are given as input a pair $(V_0, V_1)$ of linear codes in $\mathbb{F}_2^n$, and asked to decide whether $V_0, V_1$ have the same ...
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I am writing an expository essay on certain aspects of mathematical proofs, and one recurring pattern is the kind of question which is short in one direction but long in the other. A couple of ...
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I am interested in the complexity of a computational problem I encountered while studying Quran. We are given a sequence of positive integers $a_i$, we want to order them and find sums of pairs $a_{\...
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Let $C\subseteq\mathbb{F}_2^n$ be a linear code and let $P$ be the corresponding weight enumerator polynomial. That is, $$P(x)=a_nx^n+\cdots+a_1x+a_0$$ where, for $0\leq j\leq n$, we have $a_j:=\#\{v\...
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Let $T=\left[\begin{array}{cc}A&B\\C&D\end{array}\right]$ be an $(m+n)\times(m+n)$ matrix over a finite field ${\mathbb F}_{q}$, where $A$ is $m\times m$ and $D$ is $n\times n$. Consider the ...
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Let $f(x_1, \dots, x_n)$ be an $s$-sparse polynomial over a field $\mathbb{F}$, where each variable has individual degree strictly less than $d$ (i.e., $\deg_{x_i}(f) < d$ for all $i$). When we ...
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Disclaimer: this is a repost of a MS question with the same title — https://math.stackexchange.com/questions/5072398/complexity-of-the-clause-fragment-of-%c5%81ukasiewicz-logic People who know the ...
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Let $\Sigma_2 = \{0,1\}^{\mathbb{Z}}$ be the full two-shift with left-shift map $\sigma$ and the standard product metric $$d(x,y) = 2^{-\inf\{|n| : x_n \neq y_n\}}.$$ Fix $\varepsilon = 2^{-m}$ for ...
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Let $A$ be an affine subspace of $\mathbb{F}_2^n$. Let $m\leq n$ and $Q_0, Q_1$ be linear maps $\mathbb{F}_2^n\rightarrow\mathbb{F}_2^m$. Consider the following decision problem: Decide whether or not ...
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There are many NP-complete problems, e.g. SAT, CVP, SIS, graph colouring, Minesweeper etc. By definition there are polynomial time reductions from one to another of these, at least in their decision ...
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In Theorem 8 of Micciancio’s lecture notes, a reduction from the Closest Vector Problem (CVP) to its optimization version (OptCVP) is given under the assumption that the lattice basis $B \in \mathbb{Z}...
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I am interested in knowing the best complexity upper bounds for the following graph isomorphism problems (best theoretical deterministic upper bound). For some of those I already have some references (...
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A common approach to construct fast multiplication algorithms is to make an ansatz for the matrix multiplication tensor of fixed dimension and rank (e.g. $2 \times 2 \times 2$ and rank $7$ if we want ...
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An $n$ dimensional lattice is the set of integer linear combinations of $n$ linearly independent vectors in $\mathbb{R}^{d}$ ($d\le n$). The $n$ independent vectors are called the basis of the lattice,...
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I have been experimenting with a partial-distance encoding of the decision version of the Traveling Salesman Problem (TSP) using a combination of Katětov–Urysohn ideas and what I have been calling “...
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Fix distinct primes $q_1,\dots,q_t\in[2^{m-1},2^m]$ and integers $r_i\in[0,q_i-1]$ at every $i\in\{1,\dots,t\}$. Is there a way to exactly count the number of primes $a\equiv r_i\bmod q_i$ where $a\...
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I have a polynomial of degree around $2^{35}$ over a finite field $\mathbb{F}_p$, where $p$ is a 64-bit prime number. What is the most efficient algorithm for finding a root of such a polynomial? (...
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I needed to use Computational Geometry result citations for my article. The article topic is Machine Learning. Some of citations I found, but for others it appears that they belong to so called "...
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Are there known examples of naturally-occurring groups where the word problem is algorithmically solvable but not easily? I ask because I'm looking at word problems of some groups of interest to me, ...
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For $x > 0, \alpha > 0 \in \mathbb{R}$, as $x \to \infty$: $$J_\alpha(x)\sim \sqrt{\frac{2}{\pi x}}\left(\cos \left(x-\frac{\alpha\pi}{2} - \frac{\pi}{4}\right) + \mathcal{O}\left(|x|^{-1}\...
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Consider the following variant of the max-flow problem. We are given a set of flows and a network modelled by a graph. Each edge of the graph can accept only a subset of flows, which is different from ...
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Given a system of linear diophanthine equations. What is the computational complexity of checking if the system has a solution or not or finding a solution if we have an additional constraint that ...
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Let $A \in \mathbb{R}^m$ be some set with the property that it is easy (polynomial-time computation) to generate random elements $r \in A$. Is it then also easy to compute $$ P_A(x) := \arg \min_{y\in ...
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My question is Is there an algorithm in polynomial time that can find solutions $(x,y,k)$ to the Diophantine equation $$ x^2 + k y^2 = N$$ where $x,y,k$ are unknow integers, $N$ is known, but its ...
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I am interested in the complexity of this problem: Input: Given N points on integer 2D grid (N is even) Question: Can we construct an orthogonal simple polygon from the set of N mid–points? Each mid–...
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Comments on this question point out that John Gill (of the famous Baker–Gill–Solovay paper) lists a joke on his online CV: A Joke about P =?NP Gill, J., T., Ladner, R. 1973 Googling, I can’t find ...
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Assume every function is eventually nonnegative. In other words, we are interested in growth rates for measuring time complexity and such. $f = O(g)$ is equivalent to $\limsup \frac{f}{g} < \infty$,...
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Given a graph, is there a way to count the number of 'non-equivalent' obstructions to planarity to the given graph? Can this be done efficiently algebraically such as can we reduce this problem to ...
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Finding $x$ in $g^x\equiv h\bmod p$ when $p$ is prime and $g$ is generator in multipicative group $\mathbb Z_p^\times$ is the discrete logarithm problem. It is not known to be in $P$. What is the ...
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$LLL$ algorithm is vectorized version of Extended Euclidean algorithm for $\mathsf{GCD}$. Even the $m=2$ dimensions case known to Lagrange and Gauss does not have an $NC$ algorithm for shortest vector....
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Consider have the promise problem of solving for integer solutions $u,v$ in the linear system $$au+bv=c$$ where the promise is $\mathsf{GCD}(a,b)=1$. Suppose this promise problem is in $NC^1$ and the ...
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Buss defined $V_2^1$​ as a second-order bounded arithmetic corresponding to $\mathsf{PSPACE}$. Later, Skelley introduced $W_1^1$​, a third-order bounded arithmetic of $\mathsf{PSPACE}$. Since the ...
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We know there is an elementary cellular automaton (ECA) with 2 states (Rule 110) that is universal, i.e. Turing-complete. One-way cellular automata (OCA's) are a subcategory of ECA's where the next ...
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Definitions/Notation: Fix positive integers $b$ and $M$. Consider the set of real numbers which can be exactly expressed with $2M+1$ coefficients in base $b$, defined by $$\mathcal{X}(b,M):=\{x\in \...
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I am teaching algorithms and theory of computation this semester and had the opportunity to dig a bit into the details of one way functions and the P vs NP problem. This problem has resisted attacks ...
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Let $L$ be a cyclotomic field, and $P$ a prime ideal of $\mathcal{O}_L$. is there any symbol for the equation $ax^2 + by^2 =1 \mod P$ and if so, is it computable in polynomial time? if $a$ is ...
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This question was initially posted on math.stackexchange.com, but there is no appropriate answer, hence I have the right to publish it here again. Let $f(x,y) = \sum_{i = 0}^d f_i x^i y^{d-i}$ be a ...
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Given $3n$ positive numbers $a_1, \ldots, a_n$, $b_1, \ldots, b_n$, and $x_1, \ldots, x_n$, we are given a function $$f(x) = \sum_{i = 1}^n \frac{a_i}{\sqrt{(x - x_i)^2 + b_i}}.$$ Can we find all the ...
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"If we compare to non-kernel polynomial regression it is O(Tnp) where is p is dimension of polynomial while kernel polynomial is O(n^2d) + O(T*n^2) where d is original number of attributes, ...
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Let $X\subset Y$ be Banach spaces and $B_X:=\{x\in X: \|x\|_X\le1\}$ be the unit ball of $X$. The goal is to find an approximation of every element from $B_X$ with error measured in $Y$ by using at ...
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