All Questions
984 questions
2
votes
0
answers
57
views
Cylinder orientation representation
I'm trying to find an efficient computation and representation for the following problem.
Given a cylinder with height $h$ and radius $r$ with a given position $\mathbf{x} = [x, y, z]$ and $N$ number ...
- 63.9k
7
votes
2
answers
1k
views
Is a given point in the interior of the convex hull of a given finite collection of points?
Suppose I have the convex hull $P$ of a finite collection of points in $\mathbb{R}^d,$ and I want to see whether a point $p$ is contained in $P.$ This is a standard (some would say the standard linear ...
1
vote
0
answers
38
views
Fermat point amidst polygonal obstacles
Consider $k$ distinct points in 2D-plane with $n$ convex polygonal obstacles. Is there a poly-time algorithm (poly in $k$ and the total number of obstacle vertices) to find a point outside of all ...
- 1,216
2
votes
0
answers
154
views
Reduced Voronoi diagram
I am currently reading Differentiable Surface Triangulation presented at Siggraph Asia 2021.
I think most of the paper is clear to me, though I keep re-reading through to see if I miss details.
The ...
- 115
0
votes
1
answer
319
views
Correct way to conduct equilibrium scaling of linear/integer/MIP program
I would like to scale my linear/integer program and also mixed-integer program using the equilibrium scaling method. I have worked on two research papers and one research book. However, they did the ...
- 21
3
votes
0
answers
285
views
Explicit computations of finite covers of genus one curves with two points of ramification
I have an explicit genus one curve $E$ with two points $p_1$ and $p_2$ on it and am looking for an explicit degree seven cover $X\to E$ with ramification precisely over $p_i$, with a single preimage ...
- 5,186
4
votes
2
answers
314
views
Connecting $2n$ points in $\mathbb R^2$ with line segments s.t. each point belongs to exactly one line segment
I'm trying to do a certain simulation related to the toric code and I'm looking for an algorithm that connects $2n$ points ($n \in \mathbb Z_+$) in $\mathbb R^2$ with line segments with the following ...
- 5,429
1
vote
0
answers
61
views
Linear programming robustness to input perturbations
I'm running a linear program whose parametrization depends on the output of a neural network. I was wondering if there exist results on how robust linear programs are towards perturbations in their ...
- 11
1
vote
0
answers
58
views
Are cells of 4-polytopes a convex polyhedron by definition?
I'm going by the Wikipedia definition for a 4-polytope.
Do by definition, cells of 4-polytopes have to be a convex polyhedra?
If not, then are there polyhedra with non-convex faces?
If yes, is it the ...
- 11
12
votes
2
answers
11k
views
Covering a polygon with rectangles
I am trying to cover a simple concave polygon with a minimum rectangles. My rectangles can be any length, but they have maximum widths, and the polygon will never have an acute angle.
I thought about ...
- 63.9k
3
votes
0
answers
260
views
What is the VC-dimension of regular convex k-gons in the plane?
Recall the relevant definitions:
Let $H$ be a family of sets in $\mathbb{R}^d$. The intersection of $H$ with a point set $C$ is defined as $H\cap C = \{h\cap C\mid h\in H\}$. The VC-dimension of $H$ (...
- 131
2
votes
1
answer
644
views
How to maximise infinity norm of $x$ with constraint $Ax \le b$ using linear program? [closed]
I want to maximise the infinity norm of $x$, subject to constraint: $Ax \le b$. I think you can use a linear program to solve this, but how do you go about formulating it?
- 162
1
vote
0
answers
64
views
Angles between edges of a geometric graph and graph invariants
Are there any clever ways in which the angles between edges in a geometric graph are encoded in the graph spectrum, or another object associated with the graph?
I'm interested to see what else is ...
- 640
0
votes
0
answers
115
views
Explicit equation for border of the Minkowski sum of sets
Assume we have sets of the form
$$
M_j = \{x\in\mathbb{R}^d : f_j(x) \le 0,x \ge 0\}
$$
where $x\ge 0$ means $x_i \ge 0 \quad \forall i=1,\dots, d$.
Goal
I am looking for an (explicit) representation ...
- 63.9k
1
vote
0
answers
258
views
How to do an elevated 2D Delaunay triangulation?
This is what I call an elevated Delaunay triangulation:
This is also called a 2.5D Delaunay triangulation.
To do it, I simply perform an ordinary 2D Delaunay triangulation with the (x,y)-coordinates, ...
- 2,560
1
vote
0
answers
49
views
Comparing convex planar regions of equal perimeter and area - 2
We try to extend On comparing planar convex regions of equal perimeter and area .
Given two planar convex regions C1 and C2 both with unit perimeter, we define the difference between C1 and C2 as the ...
- 5,979
1
vote
1
answer
175
views
A variation on the projective Nullstellensatz
Let $V$ be a $\mathbb{C}$-vector space, and let $f_1,\dots,f_n \in S^d(V^*)$ be homogeneous polynomials of degree $d$ for which $V(f_1,\dots, f_n)=\{0\}$.
Must there exist a positive integer $k\geq d$ ...
- 148k
5
votes
1
answer
169
views
Connectedness of semialgebraic sets via CAD
I do not know whether there is a standard or some traditional ways to decide whether a semialgebraic set is connected or not.
One way I know is the cylindrical algebraic decomposition (CAD) algorithm. ...
CommunityBot
- 1
3
votes
1
answer
143
views
Finding the smallest centrally symmetric region that contains a convex planar region
Given a convex polygonal region C, how does one find/characterize the smallest zonogon (centrally symmetric convex polygon https://en.wikipedia.org/wiki/Zonogon) that contains C?
Note 1: In question ...
- 4,219
2
votes
1
answer
118
views
Minimal vertices 3D polygon fitting between inner and outer boundaries
I have a set of 3D points representing a convex hull which I define as the inner boundary. The points are then offset outwards by a selected distance (the 'error') and I define this expanded hull as ...
- 151k
1
vote
0
answers
98
views
Solution of a simple optimization problem
Let $\mathbf{U}_1$ and $\mathbf{U}_2$ be two arbitrary unitary matrices and $\mathbf{D}$ be a diagonal matrix. What is the solution of the following optimization problem?
\begin{align}
\min_{\mathbf{...
- 63.9k
0
votes
0
answers
124
views
The best unitary matrices that approximate a matrix product
Let $\mathbf{A}$ be an arbitrary $N\times N$ complex matrix. Moreover, $\mathcal{U}_1$ and $\mathcal{U}_2$ are distinct subsets of all unitary matrices. Suppose the matrices $\mathbf{U}_1$ and $\...
- 287
3
votes
2
answers
157
views
Finding a point inside a surface
I have a triangulation of a surface without boundary in $\mathbb{R}^3$.
The triangulation gives a unit normal pointing outwards for each triangle. I need to find some point in the interior of the ...
CommunityBot
- 1
1
vote
0
answers
35
views
How to chose the start vector for the MTZ variables
In the context of LP-formulations for the Traveling Salesman Problem the MTZ constraints prevent subtours via $n$ (i.e. effectively $n-1$) additional variables $$u_1=1\\2\le u_2,\,\dots ,\,u_n\le n\\ ...
- 4,713
1
vote
0
answers
72
views
A ratio to measure 'roundedness' of planar convex regions
Ref: A center of convex planar regions based on chords
The above discussion quotes the definition of 'centralness coefficient' and defines a center of a planar convex region. 1/2 is the least possible ...
- 5,979
1
vote
0
answers
124
views
A center of convex planar regions based on chords
This is based on Chapter 6 of 'Convex figures' by Yaglom and Boltyanskii. This post also continues On two centers of convex regions.
A point $P$ in the interior of a planar convex region $C$ divides ...
- 5,979
2
votes
1
answer
113
views
Existence of fine approximate of a convex body in $\mathbb R^d$ with convex hull of $\mathcal O(d)$ points
Let $K$ be a convex body in $\mathbb R^d$ which contains the origin and let $\theta \in (0,1)$.
Question. Is it always possible to find $n$ points $x_1,\dotsc,x_n \in \mathbb R^d$ such that
$$
\theta ...
- 10.6k
1
vote
0
answers
200
views
Drawing a 3D object in a 3D environment, and converting to math [closed]
So I have been granted a free time and I want to work on a project but first I had to research.
As we know, lines have infinite points, and with lines, we can create infinite shapes. I want to let ...
- 11
0
votes
0
answers
108
views
How to find a set given its support function
Let $\mathcal{U}$ be a convex and compact set. Its support function is defined as $\delta^*(v|\mathcal{U})=\sup_{u\in \mathcal{U}} v^T u$. Assume that we are given the support function $\delta^*(v|\...
- 53
9
votes
5
answers
13k
views
Get a point inside a polygon
I have a 2D polygon of arbitrary geometry. I need to find any point that is inside of that polygon. Taking the center won't work, because the polygon might not be convex. Is there a way to quickly ...
0
votes
0
answers
40
views
Subtour-gluing constraints for ILP formulation of TSPs
If one doesn't want to introduce additional variables to the ILP of a TSP instance, one has to add exponentially many so-called subtour-elimination constraints; in practical calculations subtour-...
- 13.2k
0
votes
0
answers
96
views
Why is Gaussian distribution always chosen for smoothed analysis?
I came across the algorithmic perfomance analysis model of smoothed analysis. In all references that I read a Gaussian distribution was used for perturbation (e.g. Spielman and Teng 2004 for the ...
- 29
1
vote
1
answer
82
views
Do we really need degree constraints for ILP formulations of TSP problems
The Dantzig-Fulkerson ILP-formulation of the symmetric TSP is
$$\min\sum\limits_{i=1}^{n-1}\sum\limits_{j=i+1}^n c_{ij}x_{\lbrace i,j\rbrace}\quad\text{s.t.}\\ \sum\limits_{j\ne i,\,j=1}^n x_{\lbrace ...
- 5,429
3
votes
2
answers
232
views
Partition of polygons into 'congruent sets of polygons'
Definition: Two finite sets of polygons $A$ and $B$ are congruent if we can match polygons in $A$ in a one-one manner with polygons in $B$ with each matched pair of polygons mutually congruent.
...
- 7,226
2
votes
1
answer
272
views
Triangulations of point sets — obtuse and acute triangles
Given a planar configuration of points in general position. It is known that the Delaunay triangulation is the 'fattest' triangulation possible. It is also easily seen that given 7 points with 6 of ...
- 5,979
1
vote
0
answers
30
views
On partitioning n-gons into pieces with reflection symmetry
Is 3(n-2) a tight lower bound on the least number of reflection symmetric pieces that any general n-gon can be cut into? What if we consider only convex n-gons?
A kite is a reflection symmetric ...
- 5,979
1
vote
0
answers
58
views
Second-order envelope theorem for linear programming
Consider parameterized linear programming $V(\theta) = \max_x \langle c(\theta),x\rangle$ s.t. $A(\theta)x\leq b(\theta)$, $x\geq 0$. Let's also assume $c,A,b$ are infinitely differentiable with ...
- 373
1
vote
1
answer
475
views
Sufficient conditions for a system of linear inequalities to admit a solution
I am looking for sufficient conditions such that a system of linear inequalities of the type $A x >0$ admits a non-negative solution $x \in \mathbb{R}^n_+$. I know a few properties of the $m \times ...
1
vote
0
answers
37
views
Sum of all integer binary solutions of a TUM linear system
I have the following problem: $A x = b$ where $A$ is a $m \times n$ total unimodular matrix (TUM) with entries in $\{0,1\}$ and $b$ is a $m$-vector of strictly positive integers. Let $\mathcal X$ be ...
- 11
2
votes
1
answer
192
views
On some optimal containers of a set of points on the 2D plane
Given a set of N points in general position on the plane, the problem is to give efficient algorithms to find
the smallest semicircular region (semidisk) that contains the points
the smallest ...
CommunityBot
- 1
4
votes
1
answer
866
views
When is a triangular matrix totally unimodular?
I have a {0,1}, invertible, triangular matrix, that I would like to show is totally unimodular. Are there any known results on the total unimodularity of classes of triangular matrices?
- 32.1k
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?
- 32.1k
16
votes
3
answers
1k
views
Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?
Let $P$ be a convex polytope in $\mathbb{R}^d$ with $n$ vertices and $f$ facets.
Let $\text{Proj}(P)$ denote the projection of $P$ into $\mathbb{R}^2$.
Can $\text{Proj}(P)$ have more than $f$ facets?
...
- 32.1k
6
votes
0
answers
219
views
How big a box can you wrap with a given polygon?
Question: Given a convex polygonal region, how does one find the box (rectangular parallelopiped) of maximum volume that can be wrapped with this region? While wrapping, if needed, some portions of ...
- 5,979
2
votes
0
answers
76
views
Polyhedron coordinate bound
Given a polyhedron
$$Ax\leq b$$
where we assume $A\in\mathbb Q^{m\times n}$ and $b\in\mathbb Q^{m}$ and it takes $L$ bits to represent the inequalities what is a good bound on the quantity $\|y\|_\...
- 13.9k
5
votes
2
answers
241
views
On intersections of several convex regions
Question: Given n convex planar regions. Required to place them (in suitable position and orientation) so that that part of the plane lying under all the regions (their common intersection) is of ...
CommunityBot
- 1
0
votes
0
answers
165
views
Minimum circumscribed ellipsoid of $\mathcal H$-polytope
Given matrix $A \in \mathbb{R}^{m \times n}$ and vector $b \in \mathbb{R}^n$, consider the $\mathcal H$-polytope $P$ defined as follows
$$ P := \left\{ x \in \mathbb{R}^n : Ax \leq b \right\} $$
I ...
0
votes
0
answers
137
views
Any technique for linearization, or linear approximation?
Consider the following Matrix constraint:
$$
\begin{bmatrix} -U+\psi\Sigma_b^{-1} & V \\ V^T & -V^TU^{-1}V+\tau_2 -\psi \end{bmatrix} \leq 0
$$
where $\Sigma_b$ is a known positive definite ...
1
vote
1
answer
464
views
Relation between variables (vertexes, edges, regions and faces) in three dimensional Voronoi diagram
A Voronoi diagram is a kind of tesselation that divided the medium into polygons in 2D and polyhedrons in 3D. In two dimensions, any Voronoi diagram has vertexes(V), edges(E) and regions(F) that equal ...
CommunityBot
- 1
12
votes
3
answers
1k
views
Is every graph an edge-crossing graph?
Consider a circular drawing of a simple (in particular, loopless) graph $G$ in which edges are drawn as straight lines inside the circle. The crossing graph for such a drawing is the simple graph ...
- 32.1k