All Questions
984 questions
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To partition a triangle into $n$ convex pieces with sum of number of sides over all pieces maximized
This post is a variant on To cut a triangle into $n$ $p$-sided polygonal regions.
Question: Given a positive integer $n$, a triangular region is to be cut into $n$ convex pieces so that the sum over ...
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28
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Integral hull of a polyhedron Q is polyhedron
Let $Q \subseteq R^n$ be a rational polyhedron and let $Q_I=Convexhull(Q \cap Z^n)$. By finite basis theorem, we have $Q=P+C$ for some rational polytope $P$ and finitely generated cone $C$ where $C=R_+...
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Is it possible to backtrack an optimization solver? [closed]
I have an optimization problem and was using a linear programming optimizer to find solutions. However, I find that past a certain size, the problem becomes "infeasible" and has no solutions....
2
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1
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Interpreting mincost flow dual variables
Consider the task of finding flow of size $b$ with minimum possible cost.
It may be formulated as linear programming in a following way:
$$\boxed{\begin{gather}
\min\limits_{f_{ij} \in \mathbb R} &...
CommunityBot
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Existence of element $(x_0,y)$ in a set of common zeros for all $(x_0,y)$ satisfying system of inequalities
Let $f_1,f_2,\cdots,f_n,g_1,g_2\cdots,g_m\in \mathbb{R}[x,y]$, then define the affine variety and semi-affine variety as follows:
$V(f_1,f_2,\cdots,f_n):=\{(x,y)\in\mathbb{C}^2: f_1(x,y)=f_2(x,y)=\...
CommunityBot
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2
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Any idea of solving an optimization problem with cubic constraints?
I have the following optimization problem with cubic constraints, which is hard to solve. Are there any ideas, or related references, of solving such a problem?
$$ \begin{array}{ll} \underset {y, z} {\...
CommunityBot
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Easy instance of set cover
I am trying to prove that a natural greedy algorithm solves the following instance of the set cover problem: for a set of elements $e\in U$ with a set of weights $w_e$, we define the cost of a subset ...
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242
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Prove that $ n \leq d+1 $ under ordering constraints in $\mathbb{R}^d$
Let $x_1, \dotsc, x_n \in \mathbb{R}^d$ and $\theta_1, \dotsc, \theta_n \in \mathbb{R}^d$ be vectors such that for every $k \in [n]$, the following inequality holds:
$$
\langle x_k, \theta_k \rangle &...
- 109k
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Inside-out dissections of solids -2
We record some general questions based on
Inside-out dissections of solids
Inside-out dissections of a cube
Can every convex polyhedral solid be inside-out dissected to a congruent polyhedral solid?...
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Reflections of Voronoi diagrams
I wonder if something similar to the following fact is known, and I would greatly appreciate any references.
Let $t_1, t_2, \ldots, t_N$ be unit vectors in $\mathbb{R}^n$.
Let $S$ denote the unit ...
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2
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Efficiently determine if convex hull contains the unit ball
Given a set of $n$ points in $\mathbb{R}^d$, is there an algorithm to determine if the convex hull contains the unit ball centered at the origin in polynomial time (in both $n$ and $d$)? The convex ...
- 221
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Computing all roots of a function with square-root terms
Given $3n$ positive numbers $a_1, \ldots, a_n$, $b_1, \ldots, b_n$, and $x_1, \ldots, x_n$, we are given a function
$$f(x) = \sum_{i = 1}^n \frac{a_i}{\sqrt{(x - x_i)^2 + b_i}}.$$
Can we find all the ...
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Constructing a minimum-volume outer approximation polytope with fewer facets
I am tackling the following problem:
Given a set of points $D \in \mathbb{R}^d$ and their convex hull, represented with $n$ facets, I want to construct a convex polytope $P$ with at most $m<n$ ...
- 63.9k
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$\mathrm{LP}$ formulation for $\mathrm{k}$-$\operatorname{opt}$ moves
$\mathrm{k}$-$\operatorname{opt}$ moves are an idea to improve non-optimal Hamilton cycles in weighted symmetric graphs by exchanging $\mathrm{k}$ tour-edges with $\mathrm{k}$ edges that do not belong ...
CommunityBot
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4
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212
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Efficient algorithm for graph problem
Let $D=(V,E)$ be a directed graph, $S,T\subset V$ and $f:V\rightarrow \{1,\ldots, k\}$ a positive, bounded weight-function and $l\in \mathbb{N}$, find a path $v_1,\ldots, v_l\in V$ with $v_1\in S$ and ...
- 63.9k
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Iterated optimal transport
Suppose we are interested in two consecutive transport plans (in the Kantorovich formulation). That is, we are given finite sets $X$, $Y$ and $Z$, endowed with probability measures $\mu_X$, $\mu_Y$ ...
- 128k
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Finding a special solution in a solution set over F2
Given a solution set of a linear system of the following form
$$
\{ \begin{bmatrix}
x_{1} \\
\vdots \\
x_{n}
\end{bmatrix} = \vec{v_1} * x_1 + \dots + \vec{...
CommunityBot
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Is there a name for a spanner graph that only considers distance to a root node?
A $t$-spanner graph of a set of points $\{p_i\}$ in the plane is a graph $G = (V, E)$ such that for any pair of vertices $p_i, p_j \in V$, the shortest path distance $d_G(p_i, p_j)$ in $G$ is at most $...
- 4,049
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635
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Tarski-Seidenberg for strict inequalities and bounded quantification
This theorem says that quantifiers over real variables can be eliminated from classical first order formulae built from equations and inequalities between polynomials with rational coefficients, ie in ...
CommunityBot
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2
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Algorithm for embedding a graph with metric constraints
Suppose I have a graph $G$ with vertex set $V$, edge set $E \subseteq {V \choose 2}$, a poistive integer $d$, and a weight function $w:E \to \mathbb{R}^{+}$. Is there a nice algorithmic way to decide ...
- 13.2k
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242
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Finding generators of symmetric cones
I have a bunch of vectors $\mathbf v_i$ in $\mathbb R^n$. I would like to consider the cone $C$ spanned by these vectors, together with all the other vectors that can be obtained by permuting the ...
CommunityBot
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58
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On partitioning the surface of a convex solid into geodesically convex equal area regions
We refer to a subset S of the surface of a convex solid C as geodesically convex if the shortest path along the surface of C joining any two points in S lies entirely within S (and if there are more ...
- 151k
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2
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213
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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|...
CommunityBot
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368
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Lot sizing problem: how to add these cuts efficiently
Consider the set of constraints of the uncapacitated lot sizing problem:
$$
\{(x,s,y)\in \mathbb{R}^n_+ \times \mathbb{R}^n_+ \times \mathbb{B}^n \;|\;s_{t-1}+x_t = d_t+s_t,\; x_t \le My_t,\; t=1,\...
CommunityBot
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114
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Mixed integer program and continuous Diophantine approximation
Let $n\in\mathbb{N}$ such that $n\geq 2$ and let $0<r<1$ be a real number. We wish to solve the following problem.
$$\min_{(t,(z_j)_{j=2}^n) \in \mathbb{R}\times \mathbb{Z}^{n-1}} t$$
subject to ...
CommunityBot
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396
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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 ...
CommunityBot
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Comparing partitions of a given planar convex region into pieces with equal diameter and pieces of equal width
We continue from Cutting convex regions into equal diameter and equal least width pieces. There we had asked for algorithms to partition a planar convex polygon into (1) $n$ convex pieces of equal ...
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Minimum of the maximum element frequency given the family size and the universe size
[Crossposted at math.stackexchange].
Consider families of sets $\mathcal{F}$ with size $n = |\mathcal{F}|$ and universe $U(\mathcal{F})$ with size $q = |U(\mathcal{F})|$.
I have written and solved ...
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356
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Left and right halves of convex curve
Let $S$ be a set of $n$ points in the plane in general position (no 3 on a line), $n$ even.
A halving line is a line through $2$ points of $S$ that partitions $S$ into 2 equal parts ($(n-2)/2$ points ...
- 60.5k
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Complexity of counting maximal points in query orthogonal rectangles
The problem stated in the title is the following: given an $n\times n$ binary matrix $M=\left(m_{rc}\right)$ report the number of $1$'s in a query rectangle
$[i,j]\times[h,k]$
$1\le i\lt j\le n,\, 1\...
- 13.2k
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Calculating an optimal scaling factor for Delaunay triangulations
consider a finite set $\mathcal{P}(x,y)=\lbrace(x_1,y_1),\dots,\,(x_n,y_n)\rbrace$ of points in the Euclidean plane and let $\mathrm{DT}(x,y)$ be the Delaunay triangulation of $\mathcal{P}(x,y)$
...
- 13.2k
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How to find a closest integer point to the intersection of two lines?
Here's a question that originates from StackOverflow.
Given are two lines on a plane, specified by equations ($a x + b y = c$) with integer coefficients. The lines aren't parallel and they don't ...
- 221
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A question on a quantitative form of Farkas' lemma
Suppose A is an $m \times n$ matrix whose entries are non-negative integers and $\mathbf{b}$ is a vector with rational entries. A version of Farkas lemma implies that if the equation $$A\mathbf{x}=\...
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Placing triangles around a central triangle: Optimal Strategy?
This question has gone for a while without an answer on MSE (despite a bounty that came and went) so I am now cross-posting it here, on MO, in the hope that someone may have an idea about how to ...
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Why is the Vietoris–Rips complex $\operatorname{VR}(S, \epsilon)$ a subset of the Čech complex $\operatorname{Čech}(S, \epsilon\sqrt{2})$?
$\DeclareMathOperator\Cech{Čech}\DeclareMathOperator\VR{VR}$I am reading Fasy, Lecci, Rinaldo, Wasserman, Balakrishnan, and Singh - Confidence sets for persistence diagrams (see here for a version of ...
- 12.9k
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Detecting a PL sphere and decompositions
Question 1. A PL $n$-sphere is a pure simplicial complex that is a triangulation of the $n$-sphere. I'm looking for a computer algorithm that gives a Yes/No answer to whether a given simplicial ...
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Algorithm to find largest planar section of a convex polyhedral solid
We add a bit more on shadows and planar sections following On a pair of solids with both corresponding maximal planar sections and shadows having equal area . We consider only polyhedrons.
Given a ...
- 151k
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2
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Polyline averaging
I'm trying to find a method that can take a collection of polylines, each described by a list of connected points on a plane, and find an "average" path through them. The input lines do not loop.
...
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An algorithm to arrange max number of copies of a polygon around and touching another polygon
A related post: To place copies of a planar convex region such that number of 'contacts' among them is maximized
Basic question: Given two convex polygonal regions P and Q, to arrange the max ...
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475
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Closest vertices of an AABB to a ray in n-dimensions
I came across this computational geometry problem and have not been able to find a satisfactory solution for it. A ray is known to originate from within an n-dimensional hypercube (AABB) in any ...
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115
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Software for computing polytopes
As can be inferred from the title, I want to do some computation on the facets representation of the polytopes given the vertices. My advisor recommended me Polymake, which is indeed useful even with ...
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Explicit computation of Čech-cohomology of coherent sheaves on $\mathbb{P}^n_A$
$\newcommand{\proj}[1]{\operatorname{proj}(#1)}
\newcommand{\PSP}{\mathbb{P}}$These days I noticed the following result of (constructive) commutative algebra, which I think is probably well known ...
- 12.9k
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Efficient counting of integer solutions to linear system
In my research, I have a particular 18x18 matrix $\mathbf{A}$ which defines the linear system $\mathbf{A}\cdot \mathbf{x} \leq \mathbf{-1}$ over the nonnegative integers. And I'm interested in ...
- 5,429
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Inside-out dissections of a cube
Ref:
Inside-out polygonal dissections
Inside-out dissections of solids
Definitions: A polygon P has an inside-out dissection into another polygon P' if P′ is congruent to P, and the perimeter of P ...
- 5,979
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83
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Practical way of computing bitangent lines of a quartic (using computers)
Are there known practical algorithms or methods to calculate the bitangent lines of a quartic defined by $f(u,v,t)=0$ in terms of the 15 coefficients? Theoretically you can set up $f(u,v,-au-bv)=(k_0u^...
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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....
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An algorithm to decide whether a convex polygon can be cut into 2 mutually congruent pieces
This post is based on the answer to this question: A claim on partitioning a convex planar region into congruent pieces
A perfect congruent partition of a planar region is a partition of it with no ...
- 5,979
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3
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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 ...
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268
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Is the maximal packing density of identical circles in a circle always an algebraic number?
There is a lot of interest in the maximal density of equal circle packing in a circle. And I thought that knowing whether or not the solution is always algebraic or not would be useful.
My original ...
- 2,858
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0
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105
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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 ...
- 63.9k