All Questions
984 questions
10
votes
2
answers
3k
views
How do you tell if a system of linear inequalities has a solution?
A naive solution would be to optimize a dummy variable via linear programming and see if a result is returned. I imagine there must be a more direct way.
- 693
0
votes
1
answer
141
views
Known graph/surface invariants that can be extracted from homology over different fields
The $Z_2$-homology of a surface viewed as a simplicial complex allows us to extract interesting invariants from the resulting homology groups. $\beta_0$ is the number of connected components, $\beta_1$...
- 4,462
10
votes
3
answers
755
views
Degree of generators of irreducible components
Let $V$ be a Zariski-closed subset of $\mathbb{A}^n_k$, where $k$ is an algebraically closed field. Assume that $V$ may be defined by polynomials of degree at most $d$ (or to put it otherwise $V$ is ...
- 4,132
3
votes
0
answers
3k
views
0,1 solution to system of linear integer equations
I have the following problem:
$A x = b$
where $A, b$ - $m \times n$-matrix and $m$-vector of nonnegative integers (respectively).
$x \in \{0,1\}^n $ - vector of binary variables, which need to be ...
- 149
1
vote
1
answer
3k
views
Covering an arbitrary polygon with minimum number of squares
I have a problem whereby, given an arbitrary polygon with any number of points, I need to cover the whole area by a number of fixed size squares. I can easily find a set of squares which covers the ...
- 51
3
votes
0
answers
220
views
Could SVD be used to optimize the partial inner-products?
Suppose a set $N$ of $n$ distinct points in $m-$dimensional space is given in $X\in\mathbb{R}^{n\times m}$. Also, suppose a subset $L\subset N$, $|L|=l<m<n$, with
$m-$dimensional coordinates in ...
- 131
4
votes
1
answer
400
views
Finding the "top" or "bottom" vertex of a simplex
A vertex $v$ of a simplex in $\mathbb{R}^n$ is a "top" vertex if there exists a point $p \neq v$ in the simplex with $v \ge p$ (i.e. $v_1 \ge p_1$, ... , $v_n \ge p_n$). Similarly, $v$ is a "bottom" ...
- 693
1
vote
1
answer
573
views
'Reference' request: Program to work with cyclic quotient singularities.
I'm looking for program code to deal with cyclic quotient singularities on normal surfaces. In particular, at the moment I need that given a singularity $p$ like $p=\frac{1}{n}(1,a)$ the code computes:...
- 2,215
7
votes
1
answer
360
views
Does the Hirsch conjecture hold for $n < 2d$?
The Hirsch conjecture asserts that the graph (i.e. $1$-skeleton) of a $d$-dimensional convex polytope with $n$ facets has diameter at most $n - d$.
After being open for decades, Francisco Santos has ...
- 7,857
9
votes
3
answers
3k
views
Complexity of matching red and blue points in the plane.
I'm just asking because I'm curious.
I was seeking references on the following problem, that a friend exposed to me last holidays :
Problem
Given $n$ red points and $n$ blue points in the plane in ...
- 4,123
2
votes
2
answers
2k
views
Solving a system of equations/inequalities that have trigonometric functions on the left-hand side
Is there any known (symbolic) method that solves a system of equations/inequalities that have trigonometric functions on the left-hand side of the system?
Ex) Find $x,y,\theta \in \mathbb{R}$ that ...
- 23
4
votes
2
answers
1k
views
Area of a Convex Polygon (Described via Half-Planes)
The intersection of a collection of half-planes describes a convex polygon, whose vertices can be constructed in $O(n \log n)$ time using a divide-and-conquer approach (e.g., intersect each half-plane ...
- 3,059
12
votes
1
answer
5k
views
Closest 3D rotation matrix in the Frobenius norm sense
Given a 3 by 3 matrix $M$ I would like to find the rotation matrix $R$ minimizing the Frobenius norm:
\begin{equation}
\|R-M\|_F
\end{equation}
Is there a closed form solution for $R$, or is it ...
- 561
1
vote
1
answer
240
views
Conditions for differentiability of minima and minimizers of linear functionals?
Let $B$ be a (real) Banach space and $C$ be a compact convex subset of $B$.
For every continuous linear functional $F$ on $B$, define
$V(F)=min_{c\epsilon C} F(c)$ and
$S(F)= { \lbrace c \epsilon C :...
- 13
4
votes
1
answer
966
views
Solving for Hamiltonian path with constraints on allowable routes through vertices
Suppose you have a complete graph with N vertexes, with a distinguished vertex $n=1$ ("start"), and you wish to find a route traveling exactly once through each vertex so that the distance along the ...
0
votes
1
answer
695
views
Minimum distance between two data sets
Suppose we have two sets of data, $X$ and $Y$, each of which contains
$10$ positive numbers. Now let us order the data sets $X=\left\{ x_{1},\cdots,x_{10}\right\}$,
$x_{1}\ge\cdots\ge x_{10}>0$ and ...
1
vote
0
answers
1k
views
How to solve simple bilinear equations under extra linear constraints
Hello,
This is the full version of a question I asked earlier. I am trying to understand whether finding a solution to the following bilinear system is computationally hard or easy:
$\lambda_i^T u_{...
- 53
1
vote
1
answer
2k
views
Can one efficiently optimize over the inverse of matrix?
Hello,
I have the following problem:
Find a non-negative matrix $L$ (i.e. $L_{i,j} \geq 0$ for all $i,j$), $L \neq I$ so that $A(I-L)^{-1}y \geq 0$ (the inequality must hold for each component), ...
- 53
2
votes
0
answers
362
views
Rational integer solutions of a linear Diophantine equation of cyclotomic integers
I am working with lattices in $\mathbb{C}$, and I want to know whether a certain vector is an element of the lattice.
In particular, suppose my lattice vectors are $a$ and $b$ and I want to know ...
- 86
6
votes
3
answers
289
views
Minimum separating subdivision in Plane
Hi
I was thinking about the following problem:
Given a planar Graph embedded in the plane and a set of points $P$ contained in the faces (no face contains more than one point) I want to determine ...
- 239
17
votes
1
answer
2k
views
How to efficiently vacuum the house
Let $P$ be a polygon (perhaps with no acute angles inside) and let $L$ be a line segment. The segment may move through the area inside $P$ in straight lines, orthogonal to $L$, or it may pivot on any ...
0
votes
0
answers
118
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
1
vote
1
answer
639
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 $S$ for ...
1
vote
1
answer
254
views
Characteristics of locally triangle-free graph
Hi
I am given a triangulation $T $ of a set of points $S $ in the plane and a disk $D$ which doesn't contain any triangle. If I now look at the subgraph $G(V,E)$ of $T $ whose vertices are the points ...
- 239
2
votes
1
answer
130
views
Are there intuitive/classically algorithmic analogues to Semidefinite programs on networks?
Many network optimization algorithms, including shortest path, push-relabel, augmenting path, etc, actually have an interpretation in terms of linear programming.
A famous application of semidefinite ...
- 2,383
0
votes
1
answer
409
views
Need help to find an efficient algorithm for the following problem!
Consider $x$ an n-dimensional vector with $x_i$ is integer in the range $[0 \dots k], k\in N$.
Given $A_{n\times n}$ is the covariance matrix of $x$.
$u$ is a given n-dimensional vector of real ...
- 131
2
votes
2
answers
418
views
Lovasz theta function - uses
Lovasz theta function bounds the Shannon capacity of graphs. What are some other uses of the function - especially in asymptotic coding theory and optimization problems?
3
votes
1
answer
397
views
Partially optimal solutions in integer linear programming
Linear programs with a totally unimodular system matrix are known to have an optimal integer point. They are therefore solvable via relaxing the integer constraints to intervals.
An other interesting ...
- 567
1
vote
1
answer
272
views
Euclidean neighborhoods on Polyhedral surface
Let $(X, Vertex(X))$ be a Polyhedral surface (defined like in Polthier) , $x_0 \in X$ a vertex. Let $B_\epsilon(x_0)$ the euclidean ball centred at $x_0$ with radius $\epsilon$, $\epsilon > max ...
- 43
5
votes
3
answers
1k
views
Algorithm for the intersection of a vector subspace with a cone of non-negative vectors
Hi,
I would like to know whether there is some more effective way of how to compute an intersection of a vector subspace of $\mathbb{R}^{n}$ with a cone of vectors with non-negative entries than the ...
2
votes
0
answers
917
views
Guessing game with guess cost
This is a question about Problem 328 on the website Project Euler. A description of the problem is provided in the previous link. I was wondering if there has been any research done on this question. ...
- 4,952
13
votes
2
answers
664
views
Complexity of a weirdo two-dimensional sorting problem
Please forgive me if this is easy for some reason.
Suppose given $S$, a set of $n^2$ points in $\mathbb{R}^2$.
I want to choose a bijective map $f$ from $S$ to the set of lattice points in $\lbrace ...
- 19.2k
3
votes
1
answer
1k
views
Graph drawing: unrooted undirected tree graphs with specified edge lengths
Has Joseph Felsenstein's equal daylight layout been analyzed by the graph drawing community? The following description is taken from his
drawtree
documentation (Wayback Machine):
"This ...
- 266
10
votes
1
answer
2k
views
Can taking the projective closure of an affine variety increase the degrees of its ideal generators?
Say we have some equations $f_1(x)=0, \ldots f_k(x)=0$ defining a variety $X$ in ${\mathbb C}^n$ (not necessarily a minimal number of generators, and not necessarily of minimal degree), and suppose we ...
- 11.2k
16
votes
2
answers
5k
views
Weighted area of a Voronoi cell
Let $X = \{ x_1,\dots,x_n\} $ denote a set of $n$ points in the unit square $S = [0,1]\times[0,1]$, and let $w = \{w_1,\dots,w_n\}$ denote a set of weights corresponding to the $n$ points in $X$. ...
- 195
3
votes
0
answers
312
views
Linear complementarity problem: principal pivoting algorithm
I'm trying to implement the "Dantzig; van de Panne and Whinston" principal pivoting algorithm for solving symmetric positive semi-definite LCPs from "The Linear Complementarity Problem" book (...
- 151
3
votes
1
answer
357
views
Mathematical Programming with other Algebras than Linear
Linear Programming is strongly entwined with linear algebra, as are many of its generalizations under the heading of mathematical programming / convex optimization.
What analogies are there for ...
- 2,383
1
vote
0
answers
204
views
Which rational subfields are corresponding to the two dimensional subspaces of holomorphic differentials
I implemented the algorithm that Felipe Voloch's suggested in his reply to the question:
Subfields of a function field
the algorithm is here:
Subfields of a function field
I considered the ...
- 601
21
votes
8
answers
4k
views
Determine if circle is covered by some set of other circles
Suppose you have a set of circles $\mathcal{C} = \{ C_1, \ldots, C_n \}$ each with a fixed radius $r$ but varying centre coordinates. Next, you are given a new circle $C_{n+1}$ with the same radius $r$...
- 311
5
votes
3
answers
8k
views
Linear programming - uniqueness of optimal solution
Is it possible to build such an objective function for a given set of constraints, so that there will be only one optimal solution?
My general problem is to get any vertex of a polytope formed by a ...
- 85
11
votes
3
answers
6k
views
Random Sampling a linearly constrained region in n-dimensions...
Hi,
So here is my problem:
Given a nonlinear, discontinous, cost function $f(x_1,x_2,..,x_N)$ along with linear constraints $x_n \ge 0, \forall n$
$x_n \le c_n$
and $\sum_{n=1}^N x_n = 1$ find an ...
- 113
2
votes
0
answers
215
views
Number of breakpoints in parametric maximum flow problems
The parametric maximum flow problem can be formulated as
$$f(\lambda) = \min_{x\in\{0,1\}^n} \left( \sum_{i}(a_i + b_i\lambda)x_i + \sum_{i,j}c_{ij}x_ix_j \right),
$$
where all $c_{ij}<0$ (so that ...
- 567
8
votes
2
answers
361
views
Computational methods for dealing with geometrically complicated solid boundaries in fluid-air interface problems
Hello,
I am a PhD student who does not have extensive computational experience seeking advice from those experienced with computational modelling as to which method would be most appropriate for ...
- 147
3
votes
2
answers
5k
views
Linear program to maximize the minimum absolute value of linear functions ?
I'd like to compute
$\max_{x,t} t$ such that $\forall i$, $t < a_i + |x - b_i|$.
where $a_i,\ldots, a_n$ and $b_1,\ldots,b_n$ are fixed and $x \in [0,1]$.
Can this be solved with a linear ...
- 500
1
vote
2
answers
407
views
Which data structure should I use for hierarchical T-meshes and PHT-splines?
Recently, I'm working on something about polynomial splines over hierarchical T-meshes, which is basically a rectangular grid that allows T-junctions. I want to do some numerical experiments but I don'...
- 163
2
votes
1
answer
823
views
Sequence of polygons containing the shortest path
Hello all,
I’m looking at the weighted region problem i.e. trying to find the shortest weighted path across a polygon subdivision, but at this point in my work, I only need to know the sequence of ...
- 23
6
votes
6
answers
3k
views
Circumference of Convex Shapes
Here is a puzzle I found in Mitteilungen der DMV (roughly, "Letters of the German Society of Mathematicians"), issue 19/2011. It was posed by Alfred Schreiber in "Wie man Hasen fangt" (How to catch ...
5
votes
0
answers
204
views
A polytope associated with the Hadamard Transform
In an investigation of whether or not a subset $V$ of "Hamming Space" $M_n = \mathbb{F}_2^n$ is a tile (i.e. whether $M_n$ can be written as a disjoint partition of translates of $V$) in http://arxiv....
- 4,636
8
votes
0
answers
1k
views
Infinite Linear Programming
I'm trying to prove optimality for a continuous linear program. That is, I have a linear program with an uncountable number of variables and constraints. I'm not sure how to demonstrate feasibility ...
- 133
7
votes
2
answers
3k
views
Conic hulls and cones
Suppose I have a number of vectors in $\mathbb{R}^n.$ The first question is: what is the most efficient algorithm to compute their "conic hull" (the minimal convex cone which contains them)? The next ...
- 96.4k