All Questions
2,368 questions
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
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
0
votes
0
answers
99
views
Perfectly balanced spanning trees
I call a spanning tree perfectly balanced if, after a two-coloring of the tree-graph's vertices
the two vertex sets that are defined by the assigned colors have equal cardinality and
the two vertex ...
- 13.2k
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 ...
- 235
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 ...
0
votes
0
answers
52
views
What do optimal tours tell about finite point sets?
Let $T_{\mathrm{MIN}}$ and $T_{\mathrm{MAX}}$ denote the shortest, resp. longest Hamilton cycle through a set of $n=2k+1$ points.
Let further $S_{\mathrm{MIN}}$ and $S_{\mathrm{MAX}}$ be the "...
- 13.2k
0
votes
0
answers
47
views
Cutting convex regions into equal diameter and equal least width pieces - 3
We add a bit to Cutting convex regions into equal diameter and equal least width pieces - 2. There, we asked, for example: If we divide a 2D convex region C into n convex pieces such that the maximum ...
- 5,979
0
votes
1
answer
116
views
Iterations of Dantzig-Wolfe Decomposition for a Simple Linear Programming problem
This arises from an engineering problem I am working on. Let $\mathbf{c}_i,\mathbf{a}_i,\mathbf{b}_i\in \mathbb{R}^{d}$ be a given set (collection) of vectors where $i\in\{1,\dots,n\}$. Define the ...
- 1,421
0
votes
0
answers
98
views
Asymptotic optimal sphericity
How quickly does maximum sphericity of polyhedra with $n$ faces approach 1 as $n→∞$? I can show that sphericity $1 - \frac{5 \sqrt{3} π}{27n} - O(n^{-3/2})$ is possible. Is this, especially $O(n^{-3/...
- 7,545
0
votes
0
answers
93
views
Number of vertices in a polyhedron
Consider polytopes
$$A_1[x_{1,1},\dots,x_{1,m_1},z_{1}]'\leq b_1$$
$$A_2[x_{2,1},\dots,x_{2,m_2},z_{2}]'\leq b_2$$
$$B[z_{1},z_{2},z]'\leq c$$
having vertex count $v_1,v_2$ and $v$ respectively.
We ...
- 13.9k
0
votes
0
answers
134
views
Intersection of descending series in a free group
I have stumbled upon a problem. It can be stated in the following way: Let $E$ be a finitely generated free group. Denote $\gamma_n(E)$ the $n$-th term of the lower central series. Consider a ...
0
votes
1
answer
65
views
Growth of number of isomorphism types of automorphism groups of convex 3-dimensional polytopes
To formulate my question precisely: let $s_k$ be the number of isomorphism types of automorphism groups of convex 3-dimensional polytopes with $k$ faces. Are there any references discussing the ...
- 123
0
votes
0
answers
90
views
On Covering a Planar Region with Copies of a Tile of Different Shape
Background: Consider trying to cover the largest possible scaled copy of a planar region $C$ with specified shape with n instances of a tile $T$ of specified shape and size. Several families of this ...
- 5,979
0
votes
0
answers
134
views
Spread of a disease on a modular chessboard (torus) - lower bound
I learned about the following result from one of Peter Winkler's books:
It is impossible to infect the entire $n\times n$ chessboard (usual chessboard) starting from fewer than $n$ infected cells.
The ...
- 1
0
votes
0
answers
68
views
Convex optimization under asymmetric loss in infinite dimensional space
The following problem is common in financial economics
$$ \min_{m \in L^2} \mathbb{E}[ \phi(y(\theta)-m)] \quad \text{s.t. } \mathbb{E}[ mx ]= q $$
That is, given a random variable $y(\theta)$ ($\...
- 45
0
votes
0
answers
36
views
Vertex configuration to tile repeat unit
I am working with elongated triangular tiling which has a vertex configuration of 3.3.3.4.4 and noticed that the representative symmetry is not the repeat unit. Is there a general formula to convert ...
- 1
0
votes
0
answers
108
views
Solutions to matrix equations in the non-negative integers
For an integer matrix $S$, and an integer vector $y$, I'm looking for solutions to $xS = y$ where the entries in $x$ are in the non-negative integers.
I've been doing this with Sage's mixed integer ...
- 101
0
votes
0
answers
43
views
Is there a reversible fully polynomial-time approximation scheme for polygonal billiards?
Let $P \subset \mathbb{R}^2$ be a polygon with rational coordinates, and consider discrete billiards inside $P$, where a ball (of zero radius) moves by steps of fixed length on each step, in a ...
- 6,652
0
votes
1
answer
139
views
Linear programming with exponential inequalities and rational variables
If we are given a set of real linear inequalities then using elimination theory or just linear programming we can decide. If the program also has inequalities of form $2^x\leq g$ in addition to linear ...
- 1,826
0
votes
0
answers
151
views
How many Shapes are possible to create using Voxels?
Let's suppose I have Big Cube of x cm by y cm by z cm, simmilar to this one:
This big cube is made of tiny little cubes of t cm.
All of this little cubes are transparent, but some of them are red
...
- 101
0
votes
0
answers
65
views
Slices vs harmonic functions — intuition/definitions question
One can identify functions on slices of the boolean hypercube $[n] \choose d$ with harmonic multilinear functions of degree at most $d$.
I don't really understand the intuition behind the harmonic ...
- 340
0
votes
1
answer
93
views
Rule to determine rotationally invariant orders of the points of arbitrary 2d splines
I would like to find a rule to determine the order of the points of arbitrary 2d splines, which should be invariant with respect to rotation (as far as possible).
To illustrate the problem, let us ...
- 103
0
votes
0
answers
43
views
Minimizing along independent directions, nonlinear programming
Good afternoon, I am studying the book Nonlinear Programming: Theory and Algorithms (by Mokhtar S. Bazaraa, Hanif D. Sherali, C. M.) particularly the Theorem $7.3.5$. I'm not sure I understand this ...
- 193
0
votes
0
answers
101
views
How can we analytically solve this max-sum-min problem?
Let $I$ be a finite set, and $A_{ij},B_{ij},x_i,y_j\ge0$. I want to find the choice of $x_i,y_j$ maximizing $$\sum_{i\in I}\sum_{j\in J}A_{ij}\min\left(x_i,B_{ij}y_j\right)\tag1$$ subject to $$\sum_{i\...
- 167
0
votes
0
answers
35
views
Converting a vector in a cone statement to inequality constraints
I would like to convert the following condition for $x$
\begin{align}
x = N \lambda, \lambda \geq 0
\end{align}
to a pure linear inequality form, i.e. find an $L$ and eliminate $\lambda$
\begin{...
- 1
0
votes
0
answers
232
views
What do square roots as minimums have to do with Harmonic numbers?
In an earlier question where I conjectured (and GH from MO confirmed) that the von Mangoldt function is the limit at s=1 of a certain Dirichlet series:
$$\Lambda(m)=\lim_{s\to 1+}\zeta(s)\sum_{d\mid ...
- 1,183
0
votes
0
answers
89
views
Why there isn't lexicographically smallest solution to a bounded linear program?
I am learning computational geometry when I run into this confusion. "A bounded 2D linear program may not have a lexicographically smallest solution", as the book says. I wonder why? I think we can ...
- 1
0
votes
1
answer
490
views
Relax a rectangular linear assignment problem
I wonder if there is any literature on the following problem
$$\begin{array}{ll} \underset{X \in \mathbb R^{m\times n}}{\text{minimize}} & \displaystyle\sum_{i,j} C_{i,j} X_{i,j}\\ \text{subject ...
- 205
0
votes
0
answers
99
views
Is this Graph Iteration Already Known?
When attempting to set up an ILP formulation for a weight-minimal cubic spanning tree (i.e. one with vertex degrees either 1 or 3) I needed connectivity constraint, but misremembered the contents of ...
- 13.2k
0
votes
0
answers
46
views
linear inequalities and reference request
I have proved and am using the following simple lemma in my current research problem:
Let $\{a_1,...,a_m\}$ and $\{b_1,b_2,...,b_n\}$ be set of positive numbers such that $\sum_{i=1}^m a_i < \sum_{...
- 1,269
0
votes
0
answers
193
views
What breaks down in the theory of affine hyperplane arrangments?
It appears to me that there is a substantial amount of combinatorial algebra and geometry supporting the theory of central hyperplane arrangements (See Topics in Hyperplane Arrangements, Aguiar and ...
- 101
0
votes
0
answers
368
views
Finding a point in the relative interior of the convex hull of a set of integer-valued vectors
Let $X \subset \mathbb{Z}^n$ be the set of integer-valued vectors satisfying a system of linear constraints. We can suppose that $X$ is the set of integral points in a given polyhydral set $Y \subset \...
- 136
0
votes
0
answers
252
views
Number of Generators of the Cohomology Ring of the Grassmannians
For complex projective space, its cohomology ring has $1$ generator. Extending up to the first Grassmannian which is not a projective space, that is, Grass$(4,2)$, a direct investigation shows that it ...
- 449
0
votes
1
answer
81
views
Can convex combinations of indicator functions for pairwise non-disjoint sets unordered by inclusion dominate one another?
Let $N$ be a finite subset of the naturals. Let $P$ be a set of subsets of $N$ such that:
1) $P\neq \varnothing$,
2) $\forall x\in P, |x| >1$,
3) $\forall x,y\in P,$ if $x\neq y$, then $x\not\...
- 3
0
votes
0
answers
68
views
A seemingly easy integer programming question
Let $k, m \in \mathbb{Z}_{ > 1}$. Let $a \in \mathbb{Z}_{> 0}^m$ and $t \in \mathbb{Z}^k$. Let $\varepsilon = (\varepsilon_{i,j})_{1 \leq i \leq m \\1 \leq j \leq k}$ be a matrix with entries in ...
- 501
0
votes
0
answers
890
views
Maximum shortest path problem
I have the following problem. You have a graph and every edge has a certain set of possible weights. The question is to find the assignment of those weight which will maximize the shortest path.
In ...
- 342
0
votes
0
answers
135
views
Number of polyhedra with N faces?
A. Up to isomorphism, how many polyhedra with N faces are there? Assume each face can be a triangle, square, pentagon, hexagon, etc... Furthermore each edge can be resized to any nonzero positive ...
- 101
0
votes
0
answers
180
views
Question on solving an optimization problem using Variable splitting and ADMM
Tell me if I have found the right approach to the following optimization problem:
$$
min_{x} \frac{1}{2}\left \| Ax-b \right \|_2^2
\\
s.t. \ \ \Phi v=x \ , \ {x^T(1-x)}=0
$$
$A$ and $\Phi$ ...
- 109
0
votes
0
answers
89
views
What is the minimal number of lines needed to partition a simplex into cells of diameter at most $\epsilon$?
I am studying a problem that requires me to partition the simplex into cells using a particular family of hyperplanes. For concreteness, consider the 2-simplex. I would like to construct lines ...
- 23
0
votes
0
answers
52
views
Efficient sampling from a polytope with large number of contraints [duplicate]
As far as I know, the most popular way to sample from a polytope (in H-representation)
\begin{equation}
\mathcal{P} := \{z \in \mathbb{R}^n | (Az)_j \le b_j\; \forall j=1,2,\ldots,m\}
\end{equation}
...
- 6,853
0
votes
1
answer
270
views
Generalized assignment problem with no integrality gap
Suppose I am solving the generalized assignment problem, so that I
am given matrices $U$ and $W$ and a vector $c$ (all three of which
have, say, positive entries), and I want to solve
$$\text{...
0
votes
0
answers
230
views
Toric morphism fiber and kernel dimensions
Given a morphism between two smooth toric varieties $f: X \rightarrow Y$, is the dimension of the kernel of $\mathrm{d}f$ at any point $p \in X$ equal to the dimension of the fiber at $f(p) \in Y$?
...
- 1,719
0
votes
0
answers
63
views
Paths on Cartesian products of graphs satisfying linear constraints
Assume integers $d > r > 0$ and a connected graph $G$ with $d$ vertices. Every point on the $r$-fold Cartesian product of $G$ with itself, $G^{\square r}$, is equivalent to a dimension-$d$ non-...
- 1
0
votes
0
answers
917
views
Inverse problem with a rank-1 update
I hope you can help me out with this. I have to find the solution x to an inverse system
$$
x=A^{-1}b
$$
This inverse problem is basically a least square problem with a rank-1 update.
$$
x=[uv^{T}...
0
votes
1
answer
214
views
Generalized Sphere Kissing Problem [duplicate]
I am currently researching discrete geometry and I am in need of an upper bound on a generalized kissing number in 3-dimensions dependent upon a parameter $\eta$ which is the radii of spheres touching ...
- 1,431
0
votes
0
answers
143
views
On 'Very Movable' Geometric Configurations (Configurations with a large degree of freedom)
Let $C$ be an $(n_r, b_k)$ combinatorial configuration that admits a geometric realization in the plane. I'm interested in the maximum number of points/lines $M$ of $C$ we can place in general ...
- 93
0
votes
0
answers
104
views
Big eigenvalues of a special stochastic matrix
Given a matrix $M$ of size $n\times n,$ we write its different eigenvalues by $x_1,x_2,\ldots,x_m$ with $m\leq n$ such that $|x_1|>|x_2|>|x_3|>\cdots|x_m|,$ and call $x_2\doteq |\lambda_2|(M)....
- 105
0
votes
1
answer
2k
views
Finding linearly independent columns of a large sparse rectangular matrix
I have a problem that necessitates solving a large non-negative least-squares
problem. My matrix A is large, sparse, highly rectangular (num rows >> num cols)
and nearly binary. However, A is not ...
- 103
0
votes
0
answers
103
views
Gauss-Newton for quotient functions
I'm optimizing a function of the form
$$
\sum \frac{ \|\mathbf{f_i}(x)\|^2 }{ g_i(x)^2 + h_i(x)^2 }
$$
where $x$ is a real vector, $\mathbf{f}(x)$ is a real vector, and $g(x)$ is a scalar. My first ...
- 561