Questions tagged [approximation-algorithms]
An approximation algorithm is an algorithm that finds an approximate solution to a (typically NP-hard) problem. The quality of the algorithm is measured by how close to the actual optimum it performs. For example, it is a constant factor approximation algorithm if it always outputs a solution that is within a constant factor of the optimum. Hardness of approximation is one way to separate NP-hard problems.
12 questions from the last 365 days
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Minimizing intersections between spanning trees of graph embeddings in polynomial time
Assume I have $N$ complete graphs $G_1, G_2,...,G_N$, and consider their embeddings $E_1, E_2,...,E_N$ in $\mathbb{R}^2$. Is there a (potentially stochastic) polynomial time algorithm to construct ...
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Approximation of Hamiltonian cycles
Let's define the $\texttt{MinHalfSimpCycle}$ search problem: Given $G=(V, E)$ a complete, undirected graph with weights on the edges. We want a simple cycle in $G$ (each vertex appears in it at most ...
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Linear interpolation vs L2-projection
I'm reading the book "The Finite Element Method: Theory, Implementation, and Applications" by Larson and Bengzon. In the first chapters there are presented two methods for approximating ...
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Is it in theory possible to perform general Miller’s algorithm inversion as used with the optimal ate pairing with large trace in subexponential time?
Let’s I have the following :
2 curves $G_1$ defined on $F_p$ and $G_2$ being the $G_1$ curve’s twist defined on $F_p^2$ both having the same prime order ; a large trace ; and $F_p^{12}$ as their ...
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Chebyshev approximation via iterated weighted least squares fits
I have the task of finding a Chebyshev approximation for a time-series; I want to check different types of functions, e.g. polynomials, rational functions, harmonics, etc.
I know that the Remez ...
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Finding survivable paths with a set of vulnerable edges
Consider a graph $G=(V,E)$ and a source-destination pair $(s,t)$. A set of edges $E'\subseteq E$ are vulnerable in the sense that at most $k$ of them may fail. My problem is to find a set of $(s,t)$ ...
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A variant of Steiner tree
Consider a directed Steiner tree problem with a source node $s$ a set $T$ of terminals with the following constraint. For each node $v$ on the tree, we assign a branching value $b_v$ as follows. (1) $...
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Finding parameters of best approximating recursion
Question:
how can the initial values $\left(a[0],\,\dots,\,a[k-1]\right)$ and the coefficients $\left(c_k,\,\dots,\,c_0\right)$ be determined that solve
$\min\limits_{a[0],\dots,a[k-1]\\ c_k,\dots,...
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In the modular exponentiation, as used with the adaptive root problem, how to chose the best base that will yield as few results as possible?
Let $n,m,w\in\Bbb N$ and $\lambda\in\Bbb P$ such that $w^\lambda \mod m = n$, with the requirements:
$\lambda$ being a random large prime such as $w^\lambda > 2\times m$
$1 < n < m−1$.
m is ...
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How to know if a random natural number is a probable semiprime?
Let that $n\in\Bbb N$ generated from a hash function where $n$ is long enough to be hard to factor in the gnfs algorithm. How to check if $n$ is probably a semi‑prime in a faster way than factoring it ...
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Approximation with "quantile-constraints"
Question:
given:
$$\begin{align}&\phantom{=}\lbrace \left(x_i,y_i\right),\ x_i\in\mathbb{X}, y_i\in \mathbb{R}\rbrace_{i=1}^n\\
&\phantom{=}f: (x\in\mathbb{X},\,p_1,\dots,\, p_k\in\mathbb{R})\...
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How to recover integer part from known fractional root part?
Suppose you have $r=n+f$ where $n\in\mathbb{N}$ and $f\in (0,1)$. I know that $r^2$ is an integer and I can also get as many digits of $f$ as I like, is there a way to recover the value of $n$?
Thank ...
