The approximation-algorithms tag has no usage guidance.

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### How can I efficiently check a point lie in 4 circles on sphere? [migrated]

Given coordinate of 2 points $A,B$ (Cartisian or longitude-latitude coordinate) on sphere of radius $R_1$. Point $O$ is middle of $AB$, 2 points $E$ and $F$ is derivation from $O$ by a distance ...

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49 views

### Polynomial time approximation schemes

The relationship between the minimum vertex cover and maximum independent set is a well established one. I was wondering if
If for some class of graphs $\mathcal{S}$ there exists a PTAS or even an ...

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19 views

### Performance guarantee of RLF [closed]

I cannot manage to find the performance guarantee of the Recursive Largest First (RLF) algorithm for approximating the chromatic number of a graph.
I know DSATUR has a $\mathcal{O}(n)$ guarantee, ...

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67 views

### Group Travel Salesman Problem

For the following Group Traveling Salesman problem, I'd like to know if there exists some poly-time approximation algorithm with constant approximation factor.
Group TSP is defined as follows: Take a ...

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213 views

### State of the art in the expected length of the Longest Increasing Subsequence of a random permutation

I have been reading about the topic motivated by a problem I read that asked for the first three digits of the sum of the LIS lengths in all permutations of length $n$. It is easy to see that we are ...

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141 views

### Polynomial-time algorithm solving approximately a generalization of the travelling salesman problem

Take a graph $G$ and a number of sets of nodes of $G$. The problem is to find the shortest path passing through at least one node in each node set. If each node set consists of only one node, the ...

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32 views

### Solving a Certain Constrained Isoperimetric Approximation-Problem

This question is related to my question Differential Geometric Aspects of Rubber Bands, where I asked for a mathematical model of contracting rubber bands.
In contrast to my former question, the ...

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69 views

### Approximating a max-cut's intersection with other cuts

(This is a cross-post from the Theoretical Computer Science Stack Exchange.)
For the purposes of this question, a cut in a graph $G$ is the edge-set $\delta (S)\subseteq E(G)$ between some vertex-set ...

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60 views

### FPTAS for approximating the permanent of a matrix

My question concerns approximating permanent of an $n$-by-$n$ matrix.
Several approximation algorithms have been proposed in the literature for this purpose, whose time complexity depend on $n$ and ...

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119 views

### Estimating the growth rate of nondeterministic finite automata

Given a nondeterministic finite automaton $\mathcal{A}$ (or a regular expression, or a regular grammar), can we efficiently compute the number $|L_k(\mathcal{A})|$ of accepted words of length $k$?
...

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131 views

### Nearest Point to a real algebraic set

Suppose I have a compact bounded real algebraic (eventually: or analytic or semialgebraic or semianalytic set) $V$ in $\mathbb R^3$ and a point $x\in\mathbb R^3$ not in $V$. How much do we know about ...

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191 views

### Determining Roots of a Polynomial with Interval Estimates of Coefficients

Let $f$ be a monic univariate polynomial with real coefficients:
$$f_A(x) = x^n + a_{n-1}x^{n-1} + ... + a_{0}$$
The values of $A=(a_{n-1},...,a_0)$ are unknown, but are estimated as ...

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52 views

### Is it known whether Minimum Cost Multicut is APX-hard?

My questions is concerned with the following problem: Given an undirected graph $G = (V, E)$ and (edge costs) $c \in \mathbb{Z}^E$,
$$\min \left\{ \sum_{e \in E} c_e x_e\ \middle|\ x \in \{0,1\}^E \ ...

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501 views

### Construct the best piece-wise linear continuous function fitting given curve

How to construct the optimal piece-wise linear continuous function fitting given curve and given number of knots (optimal knots positions also must be determined by this method)?

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83 views

### Computing row-sum scaled unsigned Stirling nos. of the 1st kind

As the title suggests, I would like to compute probability vectors with elements proportional to (unsigned) Stirling numbers of the first kind by row. For easy reference, here is the Wiki page.
For ...

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240 views

### Approximating the action of the U(N) exponential map

Let's say that I have a curve in $\mathbb{C}^N$ given by the action of the unitary group:
$$x(t) = e^{Ht}x_0,~ H \in \mathfrak{u}(N),~ ||x_0||=1$$
Here, $H$ is an NxN skew-Hermitian matrix (for very ...

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32 views

### Inapproximability of logarithmic factor of independent set

The hardness result derived using PCP theorem for Independent set suggests that there exists some absolute constant $\epsilon_0$ such that for $0< \epsilon < \epsilon_0$, it is hard to ...

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57 views

### How to select a subset of points from a universal to minimize the distance from outside to inside?

Here is the detailed problem.
I have a set of N points in K-dimension space, called U, and I want select M points of them, called S. For each point p in U, we define the distance from p to S as
$$ ...

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**1**answer

57 views

### Heuristic for choosing n-vectors from n-sets

my given problem is:
choose n-vectors from n-sets (one vector from each set) so that the biggest element in the sum of the chosen vectors is minimal. Unfortunately the problem is NP-hard. So I'm ...

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248 views

### Approximation of curves

When constructing minimax (sup-norm) polynomial approximations of real-valued functions, well-known results say (roughly speaking) that optimal solutions are characterized by the fact that they have ...

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165 views

### Bivariate Function Approximation

I am working on a nonlinear control design and having difficulty in finding approximation of bivariate functions. Are there papers or methods discussing the following question:
For any bivariate ...

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**1**answer

81 views

### Convergence of Symmetric Iterative Proportional Fitting

Let $A$ denote a symmetric matrix of non-negative entries, whose rows (and columns) sum up to the all-positive vector $d := A\mathbf{1}$ with $\mathbf{1}$ the all-one-vector. Denote $D := diag(d)$ by ...

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226 views

### Efficient isomorphic subgraph matching with similarity scores

I'm a computer vision PhD student, and I'm looking for an efficient approximation to the following problem, which could end up helping in image to image matching. Failing that, pointers to relevant ...

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76 views

### Norms of B-spline coefficients

In Shumaker's book (Spline Functions: Basic Theory), we know that the $l^\infty$-norm of B-spline coefficients is bounded above and below by the $L^\infty$-norm of the spline itself. Are there similar ...

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246 views

### Approximation of a given function by rational functions

Given a function $1/\sqrt{x^2 -k^2}$ where k is a constant with a small imaginary part, how do you go about constructing a rational approximation? I am interested in the L_p (p=2 or $\infty$) norm of ...

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273 views

### Tiling a rectangle with weighted cells (min-max problem)

I have been struggling with a research problem. The problem can be formalized as follows:
Given a $n\times m$ matrix $A$ containing cells with non-negative integer values, partition it in $J$ ...

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### Sparse approximation of the inverse of a sparse matrix

Is it possible to approximate an inverse of a sparse matrix with a sparse matrix?
The problem comes up in numerical non-linear quasi-Newton optimization: given a sparse Hessian a good starting point ...

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273 views

### Estimate size of graph by taking random walks

Let $G$ be a connected, finite graph and let $v_0$ be a vertex of $G$. I'm interested in methods of estimating the number of vertices in $G$, based on local exploration only. What I have in mind is:
...

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353 views

### Practical error-estimates for (adaptive) Newton-Cotes Quadrature

I am looking for practical error estimates for Newton-Cotes Quadrature rules.
Most books on numerical methods I have found mainly deal with theoretical error bounds/estimates for the respective ...

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143 views

### What algorithms do you know for beltway reconstruction?

I've faced the beltway reconstruction problem and I've developed a simple backtrack algorithm, what algorithms do you know for this problem?
Beltway Reconstruction Problem:
Assume there is a set of ...

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171 views

### envelope function for a linear combination of gaussian distributions

Given a distribution $F$ defined as a linear combination of Gaussian distributions:
$F = \sum_{i=1}^n C_i*N(\mu_i,\sigma_i)$ with $\sum_{i=1}^n C_i = 1$
I want to find a Gaussian function $Q = ...

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### Is the $d$-dimensional Arrangement of Trees still $NP$-hard?

The $d$-dimensional Arrangement Problem for general graphs is known to be $NP$-hard since the special case $d=1$ (OLA) already is (Garey et al, [1976]). For Trees however, the one dimensional case can ...

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672 views

### iteratively (approximately) solving a sum of exponentials

I would iteratively have to solve the following equation at iteration $n$:
$C = \sum_{1 \leq i \leq n}{e^{\frac{x_i}{T}}x_i}$ for $T$.
Each iteration $i$ an unknown $x_i$ will be observed and $C$ is ...

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281 views

### Giving a general term of a recursive function, and upper bound for it

Let a constant $B \ge 1$, and let $l_1 = 0$, $b_1 = 0$ be the values of $l$ and $b$ (respectively) at time $t = 1$.
Let $l_{t+1} = l_t + 1$ if $b_i < B$, and $l_{t+1} = l_t$ otherwise
Let ...

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156 views

### Using Fourier Transform to speed up calculation of forces following an inverse square law

Suppose I have $n$ electric point charges in, say, two dimensions. Is there any algorithm (and I have a hunch that it might be related to the Fourier transform) to compute the net forces that act on ...

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63 views

### Approximation for accumulative set cover

Let $S_1,\ldots,S_m\subseteq U$ be subsets of a set $U$ of size $\lvert U\rvert=n$. Over all permutations $\pi$ of the set $\{1,\ldots,m\}$, I want to maximize the quantity
\begin{equation}
...

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163 views

### Optimize a convex hull on a 2D histogram so the selected points match a target shape

I have an image (can be 2D or 3D), and compute a 2D histogram of the image (for example, the pixel intensity and gradient along certain direction). There is a known target region $R^*$ in the image. I ...

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722 views

### Removing cycles from an undirected connected bipartite graph in a special manner

Consider an undirected connected bipartite graph (with cycles) $G = (V_1,V_2,E)$, where $V_1,V_2$ are the two node sets and $E$ is the set of edges connecting nodes in $V_1$ to those in $V_2$. We ...

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128 views

### Near-linear Function aproximation

Hi,
I have a growing, near-linear function, that has some "noise" in its linearity. Is there any solution, how to approximate this function ? I tried Neural Network, but ist not best...
Function ...

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310 views

### Enlcosing a set of ellipses within one ellipse

Hello,
Is there an algorithm that takes in a set of ellipses and gives back and ellipse that encloses the set?

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### When we use Bernstein polynomials in application

When it is preferable to use Bernstein polynomials to approximate a continuous function instead of using the only following preliminary Numerical Analysis methods: "Lagrange Polynomials", "Simple ...

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### A.G. Vitushkin's “Easily representable families of functions” - can it be generalized?

Background
In his monograph "Estimation of the complexity of the tabulation problem" (translated into English as "Theory of the Transmission and Processing of Information") Vitushkin studies ...

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158 views

### Pure greedy algorithm

I study pure greedy algorithms in different basises. I am interested in 1 one question: is there such a Riesz basis $D$ in Hilbert space and $f\in H$ such that
...

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355 views

### How to check numerical precision of my computation of Stieltjes-constants?

In a thread in MSE I proposed an older routine of mine for the efficient computation of coefficients; I use a very similar routine for the quick&dirty computation of the Stieltjes-constants.
...

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### How to make an approximation of path with polynom P(x,y)=0?

Hi. Imagine that a user draws on the canvas any path. I want to approximate this path with a path $P(x,y)=0$ where $P(x,y)$ - is unknown polynom. May be somebody can suggest an appropriate algorithm?
...

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### An MST-like problem with vertex selection

Consider a planar pointset in a rectangle, where every point has a color (an integer label).
We need to select one point of every color, so as to minimize the cost of a planar MST of selected points ...

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238 views

### Place n telescopes on a sphere in R^d to see the whole sky

Where would one put $n$ telescopes on the surface of the earth
to see the whole sky as well as possible ?
Use the cosine metric to define how well we can see in direction $x$:
$ \qquad \text{cansee}( ...

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446 views

### How to evaluate binomial coefficients efficiently and as correctly as possible?

This question is more precisely about evaluation with a computer, of a binomial coefficient of the form $ \binom{x}{m}$ where $x$ is a real number and $m$ a rational integer.
The reason why I ask is ...

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### Multiobjective semidefinite programming

Let $C$ be size $n \times n^{2}$.
Let $B$ be size $2^{g(n)} \times n^{2}$ where $g(n) > n$.
There is only one $\mathcal{1}$ per row of $C$ and remaining entries of $C$ are $\mathcal{0}$.
$B$ is ...

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656 views

### Lovász $\delta$ condition for LLL Algorithm

http://en.wikipedia.org/wiki/Lenstra%E2%80%93Lenstra%E2%80%93Lov%C3%A1sz_lattice_basis_reduction_algorithm
What is the importance of the $\delta$ parameter for LLL bases called Lovász condition?
...