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2
votes
3answers
109 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 ...
0
votes
1answer
29 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 \ ...
2
votes
2answers
118 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)?
1
vote
1answer
67 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 ...
6
votes
2answers
226 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 ...
0
votes
0answers
29 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 ...
0
votes
0answers
34 views

Any software that can symmetrize input sets?

Is there any software that contains symmetrization techniques ex. polarization, Steiner Symmetrization etc. I suppose not. Which software would you suggest for rigid transformations? Thank you
0
votes
0answers
16 views

Estimating Missing Shortest-Path-Tree Distances from Point Coordinates

Problem: given a finite set of points $\mathcal{P}:=\lbrace p_i\in\mathbb{R}^2 \mid i\in \lbrace 0,...,n\rbrace\wedge i\neq j\Leftrightarrow p_i\neq p_j\rbrace$ a function ...
0
votes
0answers
31 views

Approximation with Predefined Topology of Niveau Sets

Problem given are a finite, connected, undirected and, cycle-free graph (i.e. a "tree") $T(V,E)$, of which one of the vertices (w.l.o.g. $v_0$) is defined to be the root. a planar imbedding ...
1
vote
1answer
35 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 $$ ...
1
vote
1answer
48 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 ...
4
votes
2answers
227 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 ...
1
vote
1answer
114 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 ...
2
votes
0answers
56 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 ...
0
votes
1answer
149 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 ...
2
votes
0answers
38 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 ...
1
vote
1answer
210 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 ...
-1
votes
1answer
94 views

Recursion equation convergence to sqrt [closed]

I'm new here. I have a recursion equation $$ a[n] = \frac{n \cdot a[n-1]}{n+a[n-1]} + k $$ $$ a[1] = 1 $$ $$ k \in \mathbb{R}, k > 0 $$ and from computing and ploting graphs I found that it can ...
3
votes
0answers
221 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 $J$ ...
3
votes
2answers
819 views

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 ...
4
votes
2answers
177 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: ...
3
votes
1answer
216 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 ...
1
vote
1answer
135 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 ...
4
votes
0answers
125 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 = ...
2
votes
0answers
55 views

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 ...
3
votes
3answers
538 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 ...
0
votes
1answer
212 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 ...
3
votes
1answer
145 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 ...
0
votes
0answers
58 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} ...
1
vote
0answers
134 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 ...
2
votes
1answer
401 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 ...
1
vote
2answers
120 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 ...
0
votes
2answers
240 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?
4
votes
4answers
871 views

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 ...
2
votes
0answers
118 views

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 ...
0
votes
0answers
154 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 ...
3
votes
2answers
325 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. ...
3
votes
1answer
2k views

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? ...
1
vote
0answers
67 views

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 ...
3
votes
0answers
233 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}( ...
8
votes
0answers
401 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 ...
1
vote
0answers
222 views

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 ...
4
votes
1answer
556 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? ...
5
votes
2answers
506 views

Getting started: combinatorial optimization for computer scientists

I have a background in computer science and I am starting to work on some problems those are basically combinatorial optimization problems. I have good knowleges of graphs, *-flow algorithms and so ...
0
votes
0answers
109 views

sparsest cut always has solution

Hi! How to prove that sparsest cut always has an optimal solution which is the cut for some vertex-subset. Looks like it's should be a kind of fundamental theorem for sparsest cut. But I didn't ...
1
vote
1answer
214 views

Approximate Set Cover Problem by Rounding

Here is the simple algorithm for approximating set cover problem using rounding: Algorithm 14.1 (Set cover via LP-rounding) Find an optimal solution to the LP-relaxation. Pick all sets ...
1
vote
0answers
450 views

Representing vertices of a cube using linear combination of tensor product of smaller cubes

Let $n,N \in \mathbb{N}$ with $N \ge n^{2}$. Let $F[i] = \square[i]$ refer to the cube which has vertices from $\{-1,0,1\}^{n^{i}}$ ($n^{i}$ tuple of alphabets from $\{-1,0,1\} = \square[0] = F[0]$) ...
2
votes
2answers
498 views

biggest cube problem (given set of bricks)

Input: set of bricks, each one is made of 1x1x1 cubes glued together face to face, like tetris pieces. Problem: find the way of putting together those pieces to make a solid that contains biggest ...
5
votes
0answers
231 views

Any approximation algorithms for self-avoiding walks?

I've a graph whose edges are weighted by probabilities, perhaps all equal. I would like to compute the overall probability of traveling between vertices x and y in the graph after I delete each edge ...
4
votes
1answer
787 views

Searching for an inhomogeneous diophantine approximation algorithm

Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, what is an algorithm that will provide coprime integers $a$ ...