All Questions
984 questions
2
votes
0
answers
95
views
To find the longest circular arc that can lie inside a given convex polygon
Question: Given a convex polygonal region P, to find the longest connected subset of a circle that can lie entirely in P.
For some P, the optimal subset will be a full circle; otherwise, a single arc ...
- 5,979
1
vote
0
answers
91
views
Dissection of polygons into triangles with least number of intermediate pieces
This wiki article: https://en.wikipedia.org/wiki/Wallace%E2%80%93Bolyai%E2%80%93Gerwien_theorem shows the dissection of a square into a triangle via 4 intermediate pieces. It appears easy to form a ...
- 5,979
3
votes
0
answers
85
views
Computational complexity of exact computation of the doubling dimension
Given a finite metric space $X$, the doubling constant of $X$ is the smallest integer $k$ such that any ball of arbitrary radius $r$ can be covered by at most $k$ balls of radius $r/2$. The doubling ...
1
vote
0
answers
92
views
Can a square be cut into non-right triangles that are mutually similar and pair-wise noncongruent?
We add a bit to Tiling the plane with pair-wise non-congruent and mutually similar triangles and Cutting polygons into mutually similar and non-congruent pieces
A (non square) rectangle can obviously ...
- 5,979
1
vote
0
answers
65
views
Algorithm to generate configurations with kissing number 12
That the kissing number of a sphere in dimension 3 is 12 is well known. However, it is also known that there is a lot of empty space between the 12 spheres. I deduce (am I wrong?) that there are many ...
- 111
0
votes
1
answer
169
views
How to integrate an indicator function/constraint into the cost function of a linear program?
I have a mathematical model $P$ for which I optimize two cost functions say $F_1$ and $F_2$ subject to a set of constraints $C1$–$C10$.
In $F_2$, I want it to be included only when its expression ...
- 3
0
votes
0
answers
164
views
Inf-convolution of norm 1 and norm 2 square
The inf-convolution of the functions $f$ and $g$ defined on $\mathbb{R}^n$ is
$$
h(x)=\inf _{y \in \mathbb{R}^n} f(y)+g(x-y) .
$$
We can prove that if $f,g$ are convex functions, then $h$ is convex.
...
- 129
0
votes
1
answer
75
views
Robustness of doubling dimension to small perturbations
Let $M$ be a metric space. Then the doubling dimension of $M$, denoted $\dim M$, is defined to be the minimum value $k$ such that every ball in $M$ of radius $r$ can be covered by at most $2^k$ balls ...
3
votes
2
answers
240
views
Writing a smooth plane quartic as the vanishing of $Q_0Q_2 - Q_1^2$ for quadratic $Q_0,Q_1,Q_2$
It seems well-known that any smooth plane quartic can be written as the vanishing of $Q_0Q_2 -Q_1^2$. Is there a good way to work out these quadratic factors $Q_0,Q_1,Q_2$? For example, given the ...
- 679
17
votes
1
answer
580
views
Aperiodic monotile in $\mathbb{R}$
Motivation. Recently a group of researchers found an aperiodic monotile in $\mathbb{R}^2$, answering a long-standing question. There are many results in higher dimensions, so let's explore the lower ...
- 48.8k
1
vote
0
answers
41
views
Are there rectangles that can be cut into non-right triangles that are pair-wise similar and pair-wise non-congruent?
We generalize the questions of Can a square be cut into non-right triangles that are mutually similar and pair-wise noncongruent?
Can any rectangle be cut into some finite number of triangles that ...
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3
votes
0
answers
226
views
Algorithm to dissect a polygon into a minimum amount of rectangles, conditioned on a maximum overlap
I have the following problem, I have a problem regarding concave polygons. I want to write code to cover any polygon with a minimum amount of rectangles that are allowed to overlap and have no fixed ...
- 31
11
votes
2
answers
824
views
A quadratic $O(N)$ invariant equation for 4-index tensors
Consider an $O(N)$ invariant quadratic equation
$$
T_{ijkl}= T_{ijmn}T_{klmn}+ T_{ikmn}T_{jlmn}+ T_{ilmn}T_{jkmn},
$$
where $T_{ijkl}$ is a real, totally symmetric 4-tensor, and the indices run from 1 ...
- 679
2
votes
0
answers
147
views
Graph Laplacians, Riemannian manifolds, and object collisions
To preface this question, I am a part-time game developer and full-time optimization fiend.
I am working on object collisions at the moment and many resources I have found online are more-or-less just ...
- 21
2
votes
0
answers
73
views
Is this an actual solution for centroidal Voronoi tiling, or just a visual approximation? [closed]
For the capstone project at the end of my graduate Data Science studies in late 2022, I needed an algorithm that would converge to something close to centroidal Voronoi tiling, while tiling contiguous ...
- 121
3
votes
0
answers
105
views
Techniques for solving linear inequalities
For $n$ real variables $x_1, \ldots, x_n$, I have a bunch of inequalities of form $2 x_i > x_j + x_k$ or $2 x_i < x_j + x_k$, where $i,j,k$ are distinct. My goal is to determine whether this set ...
- 231
3
votes
1
answer
151
views
Computer program for polyhedral manifolds
Suppose I have a 3-manifold obtained via face identifications of a polyhedron (e.g. the Poincaré sphere presented as a dodecahedron with opposite faces glued). Is there a program that exists for ...
- 197
5
votes
3
answers
542
views
If a polyhedron in $\mathbb{R}^3$ has local intersections, does it also have more global intersections?
Consider a simplicial complex $K$. A piecewise linear map $f: K \to \mathbb{R}^n$ is an almost-embedding if $f(\sigma) \cap f(\tau) = \emptyset$ for any two disjoint simplices $\sigma,\tau$ in $K$.
...
- 141
1
vote
1
answer
331
views
Finding a special solution in a solution set over F2
Given a solution set of a linear system of the following form
$$
\{ \begin{bmatrix}
x_{1} \\
\vdots \\
x_{n}
\end{bmatrix} = \vec{v_1} * x_1 + \dots + \vec{...
- 51
2
votes
1
answer
132
views
Planar convex region maximizing the difference in 'orientation' between its smallest containing rectangle and largest contained rectangle
We say a rectangle has orientation $\theta$ if the vector from its center to the middle of its shortest side (parallel to the longest side) has some angle $\theta$ with X axis.
Consider a planar ...
- 5,979
1
vote
0
answers
89
views
Bounds for minimax facility location in a convex region
An earlier question: Facility location on manifolds
A possibly related earlier post: Cutting convex regions into equal diameter and equal least width pieces - 2
The minimax facility location problem ...
- 5,979
1
vote
0
answers
94
views
Linear Program Optimal Value
If $f(A,b,c)$ is the optimal value of a linear program
$\min c.x$
subject to $A.x \leq b ; x \geq 0.$
Does $f(A,b,c)$ have a piecewise polynomial/rational upper bound in $(A,b,c)$ on the domain of ...
0
votes
1
answer
396
views
What is the best way to choose initial basis when applying simplex method to an equality form of LP?
Currently I'm trying to write a practically fast LP solver for a sparse instance, which is by simplex method with LU decomposition and eta-matrix update. In the development I realized that I'm not ...
- 1
5
votes
1
answer
255
views
Counting points above lines
Consider a set $P$ of $N$ points in the unit square and a set $L$ of $N$ non-vertical lines. Can we count the number of pairs $$\{(p,\ell)\in P\times L: p\; \text{lies above}\; \ell\}$$ in time $\...
- 20.2k
0
votes
0
answers
36
views
ILPs with square constraint matrix
Given the Integer Linear Programming ($\text{ILP}$) problem
\begin{array}{ll}
\text{minimize} & c^T x \\
\text{subject to}& \mathbf{A}^T x \ge b \\
\text{where}&c,x,b\in\mathbb{N}_0^n,\\ &...
- 13.2k
0
votes
0
answers
63
views
Bounds for the Dispersal Problem in convex regions
We add a bit to: Bounds for minimax facility location in a convex region
Two earlier posts: Cutting convex regions into equal diameter and equal least width pieces - 2 and Facility location on ...
- 5,979
0
votes
1
answer
28
views
Calculating vertex potentials from optimal matchings
Question:
can the solution to the dual of a Linear Program be calculated directly from the solution of the primal Linear Program?
If yes, what are known algorithms and their bounds on complexity.
As ...
- 13.2k
1
vote
2
answers
124
views
Are there variants of Euclidean Steiner Tree problem that are known to be in P?
Question: The Euclidean Steiner Tree problem (https://en.wikipedia.org/wiki/Steiner_tree_problem) is NP hard. Are there non-trivial (constrained) variants of this question that are known to have ...
- 5,979
1
vote
1
answer
61
views
On largest convex m-gons contained in a given convex n-gon where m < n
This post is the inside-out variant of On smallest convex m-gons that contain a given n-gon where m<n
Given a convex n-gon region P, and an m less than n, how to find the max area convex m-gon Q ...
- 5,979
2
votes
1
answer
240
views
Is the problem of vertex enumeration from an H-representation of a polytope NP-hard?
According to the Wikipedia page on the issue, the vertex enumeration problem is NP-hard.
However, double description and reverse linear search are algorithms listed to solve the problem. Moreover, ...
- 123
0
votes
0
answers
26
views
Monotony of enforced subtour merging
Is it true that for a symmetric TSP instance in the sequence of edges generated by successively:
calculating the optimal 2-factor
adding cardinality constraints on the edgesets of the 2-factor's ...
- 13.2k
2
votes
0
answers
112
views
Understanding normalization algorithms
Let $R$ be a commutative and reduced ring, finitely presented over $\mathbb Z$. Let $\overline R$ be the integral closure of $R$ in its total ring of fractions. In https://arxiv.org/abs/alg-geom/...
- 211
3
votes
1
answer
271
views
Resultants and elimination theory
Consider an ideal $I = \langle f_1,\dotsc,f_n\rangle$ in the ring $k[x_1,\dotsc,x_m]$.
Define the $i$-th elimination ideal to be $I_i = I \cap k[x_{i+1},\dotsc,x_m]$.
For any two polynomials $f$ and $...
- 1,435
1
vote
1
answer
119
views
Optimization on non-convex set
Let $\Omega$ be an open bounded subset of $\mathbb{R}^2$ and $f\in L^2(\Omega)$ be a given function. Consider the optimization problem
$$\mathrm{min} \int_\Omega u(x) f(x) \,dx\,,$$
where a minimum is ...
- 11
1
vote
1
answer
73
views
Partitioning polygons into obtuse isosceles triangles
Ref:
Partitioning polygons into acute isosceles triangles
Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles
https://math.stackexchange.com/questions/1052063/...
- 5,979
2
votes
1
answer
201
views
Dispersion of a "random" subset of $[-1,1]^2$
Let $C$ be the square $[-1,1]^2$. Let $a_1,\dots,a_m$ be points chosen independently and uniformly at random from $C$. Let $d_m$ (dispersion) be the random variable $\max_{x \in C}{\min_{j \in [m]}{\|...
- 261
1
vote
1
answer
78
views
To optimally wrap convex laminae with paper
Ref: On folding a polygonal sheet, Multi-layered wrapping of polyhedra
Basic intent: to wrap a given convex planar lamina with a convex sheet of non-stretchable paper (such that every point on both ...
- 5,979
0
votes
0
answers
93
views
On smallest convex m-gons that contain a given n-gon where m<n
Given a convex n-gon region P, and an m less than n, will the least area convex m-gon Q that contains P be such that an edge of Q coincides with an edge of P (in other words Q cannot be such that P ...
- 5,979
11
votes
1
answer
410
views
Complexity of counting regions in hyperplane arrangements
Let $H_1,\ldots,H_n$ be hyperplanes in $\Bbb R^d$. Denote $\mathcal{H} :=\{H_1,\ldots,H_n\}$ and let $c(\mathcal{H})$ be the number of regions in the complement: $\Bbb R^d\setminus \bigcup H_i$.
...
- 17k
1
vote
2
answers
121
views
How to solve the optimization problem $\max_{\mathbf{w}}\sum_i\text{sign}(\mathbf{w}^T \mathbf{x}_i)$?
I am looking for an algorithm to solve the following optimization problem
$$\max_{\mathbf{w}}\sum_i\text{sign}(\mathbf{w}^T \mathbf{x}_i)$$
where $\mathbf{w}$ and each $\mathbf{x}_i\in\mathbb{R}^d$.
...
- 175
6
votes
2
answers
215
views
Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles
Definition: Let us refer to obtuse triangles with the largest angle strictly above a given cutoff value as 'strongly obtuse' - the definition is parametrized by the cutoff value. Likewise, strongly ...
- 5,979
-2
votes
1
answer
248
views
Are there any non-elementary functions that are computable?
Does a function $\mathit{f}:\mathbb{R}→\mathbb{R}$ being non-elementary (not expressible as a combination of finitely many elementary operations), imply that it is not computable?
The particular case ...
- 11
2
votes
0
answers
109
views
Description of a point cloud being "undersampled" wrt persistent homology, confidence level?
I am completely new to topological data analysis, so I apologise if this is a well-known area of persistent homology, as well as for any imprecise language.
Suppose we know completely the topological ...
- 21
1
vote
1
answer
181
views
Linear programming with "nice" matrices
Consider the following linear programming problem
\begin{array}{ll}
\text{minimize} & \mathrm 1^{\top} \mathrm x\\
\text{subject to} & v\le \mathrm A \mathrm x \le u\\
& \mathrm x \geq ...
- 650
9
votes
0
answers
289
views
Computer algebra tools for finding real dimension of an algebraic variety
I have a system of polynomial equations with the unknowns being real numbers. The set of solutions is infinite. What software can I use to compute the real dimension of the solution set?
The CAD-based ...
- 175
0
votes
1
answer
119
views
How many samples do you need to get constant dispersion?
Let $C_n$ be the hypercube $[-1,1]^n$. For $a_1,\cdots,a_s \in C_n$, define its dispersion $D(a_1,\cdots,a_s)$ as $\max_{x \in C_n}\min_{i \in [s]} \|x-a_i\|_{2}$. Let $0< \lambda < 1$ be a ...
- 261
1
vote
1
answer
69
views
$A^*$ algorithm to find shortest path when weights in my graph are the inverse of distance
Given a graph G=(V,E) where the weights on my edges are inverse of Euclidian distance between nodes, I want to know if I can use A* algorithm to find the shortest path. How I need to modify the ...
- 111
2
votes
2
answers
213
views
Bounding the length difference of two curves given the Fréchet distance between them
Given two simple, closed, convex, planar curves $C_1$ and $C_2$, let their lengths be $\ell_1$ and $\ell_2$, respectively, and their Fréchet distance be $d_f$. We are trying to bound $|\ell_1 - \ell_2|...
- 69
2
votes
0
answers
126
views
Checking existence of a non-crossing Hamiltonian path in geometric graphs
I am interested in the following computational problem. Given a geometric graph (i.e, a graph drawn in the plane so that its vertices are represented by points in general position and its edges are ...
- 1,305
26
votes
2
answers
4k
views
Why did Robertson and Seymour call their breakthrough result a "red herring"?
One of the major results in graph theory is the graph structure theorem from Robertson and Seymour
https://en.wikipedia.org/wiki/Graph_structure_theorem. It gives a deep and fundamental connection ...
- 290