Questions tagged [computational-complexity]
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.
1,368 questions
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Decision problem for finite (unordered) trees
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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Computational complexity of a preorder of commutativity conditions
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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Algorithms to count restricted injections
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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Terminology: commonly used name for an $\omega$ machine?
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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Evaluating the weight enumerator polynomial at special points
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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Understanding monomial cancellation in $f^2$ for sparse polynomials with bounded individual degree
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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Minimal finite-edit shadowing distance in the full two-shift
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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Reduce linear code minimum distance to lattice closest vector (CVP)
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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Best time complexity upper bounds for Graph Isomorphism problem of several graphs / digraphs classes of bounded degrees
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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How fast can we solve SVP using an SVP$_{\gamma}$ subroutine?
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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Complexity of number of primes in arithmetic progression under $P=NP$
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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Self-evident truths in Computational Geometry
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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Rough numerical approximation of the Bessel functions of the first kind
For $x > 0, \alpha > 0 \in \mathbb{R}$, as $x \to \infty$:
$$J_\alpha(x)\sim \sqrt{\frac{2}{\pi x}}\left(\cos \left(x-\frac{\alpha\pi}{2} - \frac{\pi}{4}\right) + \mathcal{O}\left(|x|^{-1}\...
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Computational complexity of a system of linear diophantine equations with a non zero constraint
Given a system of linear diophanthine equations. What is the computational complexity of checking if the system has a solution or not or finding a solution if we have an additional constraint that ...
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Polynomial Time Algorithm for Solving Diophantine Equation with large integer number
My question is
Is there an algorithm in polynomial time that can find solutions $(x,y,k)$ to the Diophantine equation
$$ x^2 + k y^2 = N$$
where $x,y,k$ are unknow integers, $N$ is known, but its ...
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What was Gill and Ladner’s joke about P=NP?
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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Testing planarity algebraically by reduction to counting problem reducible to determinant computation
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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Extended Euclidean in $NC^1$ to $2D$ shortest vector problem in $NC$
$LLL$ 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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Reference request: Proof theory in $W_1^1$
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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Complexity of evaluation of analytic functions
Given an analytic function $f(x)$ (say as combination of elementary functions and operators), is it possible to compute $n$ first bits of the value of the function on the whole interval $[a, b]$ ...
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References on P vs NP under various axiomatic systems
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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What is the fastest known algorithm for evaluating a homogeneous binary polynomial?
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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How computationally efficient are kernel tricks? [closed]
"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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What oracles make finding isomorphism (of finite structures) easy?
Below, all structures are finite, in a finite language, with underlying set an initial segment of the natural numbers. This has been edited to fix errors pointed out by Emil Jerabek in his answer ...
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Klondike Solitaire as an NP-complete game
I am not a mathematician. I am trying to understand if the paper "The complexity of solitaire" that shows this game is NP-complete also has a implicit assumption that a given hand can only ...
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