All Questions
984 questions
0
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119
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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
3
votes
0
answers
95
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Fast numerical integration of $\int_{[0,\:1)^d}\left|f_x(y)-g(y)\right|^p\:{\rm d}y$ for varying $x\in[0,1)^d$
Let $k\in\mathbb N$ and $y_1,\ldots,y_k\in[0,1)^d$ with $$\frac1k\sum_{i=1}^kh(y_i)\approx\int_{[0,\:1)^d}h(y)\:{\rm d}y\tag1$$ for every nice enough function $h:[0,1)^d\to\mathbb R$.
Now let $p\ge1$, ...
- 167
1
vote
1
answer
127
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Smallest trapeziums containing a given convex n-gon
Question: Given a planar convex $n$-gon $C$, to find the smallest area / smallest perimeter trapezium (trapezoid) - a convex quadrilateral with at least one pair of mutually parallel edges - that ...
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4
votes
2
answers
315
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 ...
- 269
1
vote
0
answers
111
views
On finding optimal convex planar shapes to cover a given convex planar shape
Covering a specific convex shape S with n copies of another specified convex shape S' (which may be different from S) is well studied - for example, https://erich-friedman.github.io/packing/index.html....
- 5,979
2
votes
1
answer
85
views
Example of worst case distributions for 4D convex hull
My understanding is that convex hull of n points in 4D could have O(n²) edges in the worst case. Source: https://sites.cs.ucsb.edu/~suri/cs235/ConvexHull.pdf
This same source writes
In 4D, there are ...
- 723
2
votes
1
answer
154
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Do there exist smaller simplicial models of barycentric subdivisions?
Let $S$ be a simplicial complex and let $Bary(S)$ denote its barycentric subdivision.
Of course, the geometric realizations of $S$ and $Bary(S)$ are homeomorphic.
However, one issue that arises in ...
8
votes
2
answers
1k
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Minesweeper as a linear algebra problem
I've written a computer program to generate and solve minesweeper games. Once I've eliminated the obvious mines and safe squares I look at each remaining connected setsin turn and formulate a linear ...
- 361
16
votes
3
answers
1k
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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?
...
- 163
0
votes
0
answers
145
views
Bound on solutions of $Ax \ge b$
Let $A \in \mathbb{Z}^{m \times n}, b \in \mathbb{Z}^{m \times 1}$.
One can show that if there is a solution of $Ax \ge b, x \in \mathbb{R}^n$ then there is one such that $\|x\|_{\infty} \le c (\|A\|_{...
- 439
1
vote
0
answers
34
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On partitioning convex polygonal regions in area ratio $t : (1-t)$ where $0<t<1/2$ with least length of cut
Question: Given a convex n-gon P. How can we efficiently find the partition of P into 2 pieces with areas in the some given ratio $t : (1-t)$ where $0<t<1/2$ such that the length of cut is ...
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15
votes
1
answer
616
views
Acute triangles in "obtuse" polygons?
Let $P$ be a convex polygon. Suppose every interior angle of $P$ is obtuse. Is it always the case that there exist three vertices $p, q, r$ of $P$ such that $\triangle pqr$ is acute?
I conjecture ...
- 173
0
votes
1
answer
147
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Is there a redundant constraint in linear programming? [closed]
From wikipedia:
But... Why do we need the $x\ge 0$ part? We can instead do $-x\le 0$, and thus saving a line in the definition (which is not a big deal but nevertheless nice).
(In order to do that, ...
- 3
27
votes
5
answers
2k
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Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?
Suppose we have a $(2m-1) \times (2m-1)$ matrix defined as follows:
$$\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}.$$
For example, if $m=3$, the matrix is
$$\begin{pmatrix}6 & 20 & 6& 0 ...
- 1,121
2
votes
0
answers
197
views
Is orthogonal polygon with crossings count NP-complete?
The are several NP-complete problems related to the construction of orthogonal simple polygons. Rapport showed that it is NP-complete to decide the existence of orthogonal simple polygon that passes ...
- 2,993
0
votes
0
answers
64
views
Alternatives to McCormick Envelope
I have an optimization problem for which I have the optimal solution obtained by the ILP.
However, when I introduced the McCormick Envelope to replace the product of a bi-linear term in its LP ...
- 3
5
votes
0
answers
167
views
Computing sums with linear conditions quickly
Let $f:\{1,\dotsc,N\}\to \mathbb{C}$, $\beta:\{1,\dotsc,N\}\to [0,1]$ be given by tables (or, what is basically the same, assume their values can be computed in constant time). For $0\leq \gamma_0\leq ...
- 20.2k
15
votes
1
answer
358
views
Are hyperbolic $n$-manifolds recursively enumerable?
Fixing a dimension $n \ge 4$, is the class of closed hyperbolic $n$-manifolds recursively enumerable?
Since hyperbolic manifolds are triangulable I can reformulate this in the following more explicit ...
- 3,399
25
votes
3
answers
2k
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Is the Ford-Fulkerson algorithm a tropical rational function?
The Ford-Fulkerson algorithm
Let me recall the standard scenario of flow optimization (for integer flows at least):
Let $\mathbb{N} = \left\{0,1,2,\ldots\right\}$. Consider a digraph $D$ with vertex ...
- 33.8k
3
votes
1
answer
107
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Results in computational geometry utilizing doubling dimension of a metric space
According to Wikipedia,
However, many results from classical harmonic analysis and computational geometry extend to the setting of metric spaces with doubling measures.
My question is: what are some ...
2
votes
0
answers
60
views
Biconvex Lens - an 'oriented' convex container for planar point sets
We continue On some optimal containers of a set of points on the 2D plane.
Let us define a biconvex lens as the intersection of two circular disks - not necessarily of the same radii. Such a figure ...
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2
votes
1
answer
84
views
'Constrained morphing' of planar convex regions
Morphing may be defined as a continuous transition of one shape to another. This post is about modifying planar regions continuously from one form to another under some constraints.
Qn: If $C_1$ and $...
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3
votes
0
answers
144
views
Hemisphere containing the maximum number of points scattered on a sphere
Consider a set of points $x_1, \ldots,x_n$ on $\mathbb{S}^{k-1}$ (the unit sphere in $\mathbb{R}^k$). The goal is finding the hemisphere which contains the maximum number of $x_i$'s. Basically, we ...
- 127
1
vote
1
answer
115
views
$\mathrm{LP}$ formulation for $\mathrm{k}$-$\operatorname{opt}$ moves
$\mathrm{k}$-$\operatorname{opt}$ moves are an idea to improve non-optimal Hamilton cycles in weighted symmetric graphs by exchanging $\mathrm{k}$ tour-edges with $\mathrm{k}$ edges that do not belong ...
- 13.2k
0
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0
answers
85
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Show that $\max_{P_X : X\in (0,1) } \left| \frac{\mathbb{E} [ f'(X) ]}{ \mathbb{E} [ f(X) ] } \right|$ is maximized by at most two mass points
Let $f$ be some given well-behaved function. Consider the following optimization problem overall probability distribution on $[0,1]$
\begin{align}
\max_{P_X : X\in [0,1] } \left| \frac{\mathbb{E} [ ...
- 671
3
votes
1
answer
111
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Constrained morphing of polygons
This post continues 'Constrained morphing' of planar convex regions
If an $m$-gon $P_m$ is to be morphed (altered continuously) into an $n$-gon $P_n$ with same area and perimeter, can one ...
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1
vote
1
answer
119
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Adding linear constraint to the domain
I don't know if it is a well-known problem, but I have been struggling to come up with an algorithm.
I have a set of linear constraints $Ax\le b$, $b\ge 0$ ($b$ and $A$ are given, $x$ is a variable). ...
1
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0
answers
58
views
Covering a unit square with odd number of equal area triangles - optimally
We add a bit to this post: Cutting off odd numbers of equal area triangles from a unit square
Question: Given an odd integer n, how does one cover the unit square completely with n equal area ...
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66
votes
3
answers
4k
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Reasons to prefer one large prime over another to approximate characteristic zero
Background:
In running algebraic geometry computations using software such as Macaulay2, it is often easier and faster to work over $\mathbb F_p = \mathbb Z / p\mathbb Z$ for a large prime $p$, rather ...
- 7,328
7
votes
1
answer
348
views
Finding maximal prefix of a simple curve
Let $S$ be a simple curve. I want to determine maximal prefix of $S$ contained in a unit circle. Is this possible, or has it perhaps already been solved in the past, and I am just unable to find an ...
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0
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0
answers
84
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1-degree SOS proof refutes Linear Programming
I am trying to understand Sums-of-Squares proof systems.
A degree $d$ Sums-of-Squares refutation for a set of polynomial equations $P = \{p_1(x) = 0, ..., p_m(x) = 0\}$ is defined as
$\sum_{i=1}^m g_i(...
0
votes
1
answer
51
views
What is the most dense sample for which the Crust algorithm returns an incorrect polygonal reconstruction?
The Crust algorithm by Amenta, Bern, and Eppstein computes a polygonal reconstruction of a smooth curve $C$ without boundary from a discrete set of sample points $S$. It is known that if $S$ is an a $\...
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1
vote
0
answers
78
views
Triangulation of polygons with all triangles having a common angle
Following Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles, we record another triangulation question.
Question: Given an n-vertex polygonal region ("n-...
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9
votes
2
answers
843
views
How did they come up with the MRRW bound?
Among the good asymptotic bounds in coding theory in the MRRW bound. It is obtained by using the linear programming problem of Delsarte's and providing a solution. The LP problem is
Suppose $C \...
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1
vote
0
answers
335
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Closed-form solution of a particular linear program
(Note: I asked a similar question at math.stackexchange but the present one is more precise.)
I have a linear program of the form:
$$\text{minimize} \space\space x_1 \space\space \text{subject to:}$$
$...
- 988
0
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1
answer
103
views
Constrained linear optimization problem on $C^1$
I am dealing with a problem of the form ($a<b$)
$$
\displaystyle \max_{v \in C^1([a, b])} \int_a^b v(x)~\mathrm{d}x, \quad \mathrm{s.t.} \int^b_a \big(-o'(x)v(x)-v'(x)o(x)\big)f(x)~\mathrm{d}x \...
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
1
vote
0
answers
82
views
Inside-out dissections of polygons - a generalization
Definitions (Inside-out polygonal dissections): a polygon P has an inside out dissection into P' if P′ is congruent to P and the perimeter of P becomes interior to P' and so the perimeter of P' is ...
- 5,979
0
votes
1
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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
2
votes
1
answer
61
views
Counting the number of pair of d-uplets with upper bounded distance
Consider two d-uplets $u = (u_1,...,u_d)$ and $v = (v_1, ..., v_d)$ both living in $\mathbb{N}^d$ with $d$ a positive integer. They both verify $$(*) \sum_{i=1}^d u_i = \sum_{i=1}^d v_i = k$$ with $k$ ...
- 55
0
votes
1
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538
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Method for (binary) optimization under constraints
I would like to know if there is a method to solve the Problem.
Problem:
Maximize the following function: $$f(p_{1,i},p_{2,i},\dotsc,p_{m,i})=\sum_{i=1}^{n}\begin{bmatrix}p_{1,i} & p_{2,i} & \...
- 3
0
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0
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272
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Finding the eigenvectors of a submatrix
Let $A=(a_{kl})$ be a matrix in $M_n(\mathbb{R})$ when $n$ is even. Let $B=(b_{kl})$ be the symmetric $2n$ by $2n$ matrix whose entries are given by,
$b_{k,l}=a_{kl}$ if $1\leq k,l\leq n$.
$b_{n+k,l}=...
- 4,058
1
vote
0
answers
110
views
Upper bound on the diameter of a convex lattice n-gon with a given area
Given the area $A$ of a strictly convex polygon with $n$ vertices with integer Cartesian coordinates, there are usually several non-equivalent polygons. The relationship between the area, the number ...
- 206
2
votes
1
answer
372
views
Who called Farkas' fundamental theorem a lemma?
Farkas proved his famous result (which, nowadays, is fundamental in optimization theory) in 1902 and called it Grundsatz der einfachen Ungleichung which may be translated as fundamental theorem of ...
- 16.4k
0
votes
1
answer
175
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Reference request: How to construct a diffeomorphism between point clouds
I'm interested in the following question:
Given two sets $S = \{x_1, ..., x_N\}$ and $T = \{y_1, ..., y_N\}$ each consisting of $N$ distinct points in $\mathbb{R}^n$, how can we construct a ...
- 637
2
votes
0
answers
117
views
Folding polygons into 'vessels'
This question is based on http://www.science.smith.edu/~jorourke/Papers/FoldingPP.pdf
Define an vessel as a convex polyhedron with one face removed - in other words, a vessel can be converted into a ...
- 5,979
0
votes
1
answer
93
views
How quickly can this IQP or its MILP relaxation be solved
Let $A\in\{0,1\}^{(n,n)}$ be a $n$ by $n$ boolean matrix (in particular think of an adjacency matrix of a graph), and consider the following optimization problem:
$$\begin{align*}&&\max_{P\in\{...
- 71
3
votes
0
answers
65
views
Cutting triangles into triangles with equal longest side
This post elaborates on a specific instance of Cutting convex polygons into triangles of same diameter .
Question: For any integer n, can any triangle be cut into n non-degenerate triangles all of ...
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16
votes
3
answers
652
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How to plot this fractal
I'm a graphic designer and my client has asked me to use this fractal in a design that I'm working on. As you can see, it's not a very good copy, so I'm trying to see if I can generate a high-...
- 263
1
vote
0
answers
86
views
Explicit form of boundary operators of topological cones
Let $\Omega$ be a regular, finite, $n$-dimensional CW complex with chain modules $\mathscr{C}_k$ and boundary operators $\partial_k$.
For many problems in computational geometry, a key operation is to ...
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