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.
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Record primality radii and zeroes of Zeta
Quite unexpectedly, I noticed a striking proximity between $AGM(n-r_{0}(n),n+r_{0}(n))$ (where $r_{0}(n):=\inf\{r\ge 0, (n-r,n+r)\in\mathbb{P}^{2}\}$ and $n$ is such that $m<n$ implies $r_{0}(m)<...
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Comparing two adjacency matrices for graph equality
I'm currently working on a project that partially involves graphs. One of the problems I'm tackling is determining whether two given matrices represent the same connected undirected graph.
So given ...
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LLL algorithm for $\mathbb{Z}[X]$-lattices
I understand the LLL algorithm for finding approximate shortest vector in $\mathbb{Z}$-lattices (where the norm function is either $\ell_\infty$ or $\ell_2$), as well as finding the shortest vector in ...
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Algorithm for optimal set partition
Given a function $f:2^{[n]} \to [0, \infty)$, the problem is to find a partition $\mathscr{P}$ of $[n]$ which maximizes $\sum_{S \in \mathscr{P}} f(S)$. I am also interested in the case where $\...
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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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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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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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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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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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How expressive is $e^A$ in the sense of universal approximation?
For any real matrix $B\in\mathbb{R}^{n\times n},n\ge 2$ and precision $\varepsilon$, is there a real matrix $A\in\mathbb{R}^{n\times n}$ such that $\|e^A-B\|_F<\varepsilon$? ($F$ refers to ...
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Discrete approximation of continuous determinantal point processes
(throughout, "DPP" denotes "Determinantal Point Process")
TL;DR: Discrete DPPs are straightforward to compute with, continuous DPPs less so. Can we approximate continuous DPPs well ...
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Approximate solution problem of rank-one modification matrix secular equation
In Golub's paper , page 327,the eigenvalues of a rank-one modification of a $n\times n$ symmetric matrix can be computed by findng the zeros of the secular equation
\begin{equation*}
w(\lambda_j)=...
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Educated guess for algebraic approximation
I found a very neat ancient hindi formula for approximating square roots using rational numbers. After doing some algebra on the formula, i came across with this recursive relation:
Given any number $...
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Lagrange's interpolating polynomial
Let $f:[a,b]\rightarrow R$ be a function that is not $C^{(n+1)}$ on $[a,b]$ but its $n$-th derivative is a Lipschitz function? How does the Lagrange's interpolating polynomial formula change? How does ...
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Approximation with special partitions of unity
Question:
what can be recommended for calculating $f(x)$ that solves $\frac{f(x)}{f(x)+f(1-x)}\approx g(x)$ for $x\in[0,1]$?
I have tried comparing Taylor series, but they look intimidating and I ...
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Choosing $k$ different assignments of binary variables in order to capture the largest volume of the joint probability distribution
Assume you have $n$ independent binary variables $\{x_1,\dots,x_n\}$ and for each variable $x_i$ you know that its value is equal to $1$ with a probability $p_i$. I would like to enumerate the joint ...
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Can we talk about approximation when the decision problem for solution existence is NP-Hard
I am wishing to design an approximation algorithm for an optimization problem where the existence of solution for corresponding decision problem is not guaranteed. Is it wise to find an approximation ...
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What is an approximation algorithm in the context of NP completeness in general
In theorem 4 of Approximability of Minimum-weight Cycle Covers Bodo Manthey proves that:
Then no approximation algorithm
for $\operatorname{Min-L-DCC}$ achieves an approximation ratio of $o(n)$, ...
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How to find an optimal sequence of merging operations?
Given a set of items, each characterized by a quality $q_i\in(0,1)$. We can merge two items of quality $q_i$ and $q_j$ to a single item $k$ of quality $q_k=f(q_i,q_j)$, where $f$ is increasing in $q_i$...
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Maximize connectivity probability with a number of edges
We are given a graph $G$, whose edges are either open or closed. Initially all the edges are closed. For each edge $e$, if we choose to activate it, then after the activation, it becomes open with ...
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Steiner tree subject to edge capacity constraint
Given a network of routes modeled as a graph where each edge $e$ has a capacity $c_e$. We have a source node $s$ and a set of destination nodes $t_i$ ($1\le i\le k$). We need to transport $q_i$ ...
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Evaluate the goodness of piecewise linear approximation of a cubic Bézier [closed]
I saw that there are different algorithms for piecewise linear approximation of cubic Bézier curves out there.
Now suppose that these algorithms can be either used interchangeably or that the ...
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What is the complexity of a special multigraph edge coloring problem
Given a multigraph such that there are 0 or 2 edges connecting every two vertices, we are to color the edges of this graph so that adjacent edges receive distinct colors. It is known that we need at ...
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Approximation of integral of gaussian function over a parallelepiped
Remark: I posted this question in math stackexchange here and computer science stackexchange https://cs.stackexchange.com/ few weeks ago but obtain no answer.
Given a multi-dimensional gaussian ...
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Do higher-order splines with Lipschitz derivatives exist on finite sets?
Fix $k\in \mathbb{N}^+$ and let $E=(e_i,f_i)_{i=1}^I\subset \mathbb{R}^n\times \mathbb{R}^m$ be a non-empty finite set with $e_i\neq e_j$ whenever $i\neq j$.
If $n=m=1$ then it's easy to see that:
$$
...
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