All Questions
Tagged with combinatorial-optimization mg.metric-geometry
32 questions
0
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0
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176
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How to find a configuration of lines
In $\mathbb{R}^3$, can anyone help find a configuration of 5 lines such that the minimum of the smallest semi-axis lengths of the ellipsoid $ \mathbf{x}^T \mathbf{A} \mathbf{x} = 1 $, where $\mathbf{A}...
- 61
6
votes
2
answers
540
views
Bound on the number of unit vectors with the same pairwise inner products
I want to know the bound on the number of unit vectors $v_i$ in $\mathbb{R}^n$ such that $\langle v_i, v_j\rangle=c$ for all $i\ne j$. I know this can be upper bounded by the number of equiangular ...
- 163
3
votes
0
answers
147
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Understanding why $\frac{\phi^5}{2}$ solves this 3D optimization problem, where $\phi$ is the golden ratio
I would like to understand the deep meaning of a solution which arises from an optimization problem discussed in a paper of mine since it can be simply stated as $\frac{\phi^5}{2}$, where $\phi := \...
- 1,451
1
vote
0
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67
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Conjecture on the increasing efficiency of the shortest minimum-link polygonal chains covering any grids of the form $\{0,1,2\}^k$ as $k$ grows
From the well-known Nine dots problem, we know that we need a polygonal chain with at least $4$ edges to connect the $9$ points of the planar grid $G_{3,2}:=\{\{0, 1, 2\} \times \{0, 1, 2\}\} \subset \...
- 1,451
8
votes
1
answer
567
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Joining the $2^k$ points of $\{0,1\}^k$ with the shortest tree
Let $k$ be a given positive integer, and then consider the unit hypercube $\{0, 1\}^k \subset \mathbb{R}^k$ (i.e., a $k$-dimensional "cube" in the well-known Euclidean space).
We need to ...
- 1,451
4
votes
2
answers
150
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$O(n^{2-\epsilon})$ bound on choosing $n$ points on the hypersphere to maximize $\pm 1$ weighted sum of their $\binom{n}{2}$ inner products
Given $n,d\in\mathbb{N}, n\gg d$, I'm looking for a bound on the maximum (or minimum) expected value of the following game:
Draw a vector $\epsilon\in\{\pm 1\}^{\binom{n}{2}}$, uniformly at random. ...
- 41
6
votes
0
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74
views
Roundest polyhedra: how well can we bound the edge count of their faces?
By "roundest" I mean having the lowest surface area for the highest volume, given a fixed number of faces $n$. There've been a few questions about them on here (including from me), but I'm ...
- 3,612
2
votes
2
answers
163
views
References for geometric properties of optimal Euclidean traveling salesman tour
Consider a finite set of points $V \subseteq \mathbb{R}^2 $ as a TSP-instance under the standard $\| \cdot \|_2$ norm. (TSP stands for traveling salesman tour.) We know that every optimal TSP tour $T$ ...
- 29
1
vote
0
answers
82
views
Efficient algorithm for a distance on strings
Let $(M,d)$ be a metric space. Consider two sequences $a = (a_i)_{i=1}^n$, $b = (b_i)_{i=1}^m$, $n, m \in \mathbb{N}$ with elements in $M$. For two sequences $[n],[m]$, call $$\Gamma([n], [m]) = \big\{...
1
vote
0
answers
68
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Facility location and traveling salesman
This question is based on Distributing points evenly on a sphere and Facility location on manifolds
The 'dispersal problem' (which can be mapped to packing disks in many cases) places $n$ points in a ...
- 5,979
7
votes
1
answer
171
views
Metric TSP with integer edge cost
Given a metric TSP with integer edge cost upper-bounded by a constant $C_{\max}$, can we find an poly-time algorithm solving this TSP instance?
- 367
2
votes
0
answers
131
views
Maximum number of regions in a disk partitioned by pairs of parallel chords
We are given a disk $D$ in $\mathbb{R}^2$. Let $C$ be its boundary (i.e., the circle bounding $D$ on its plane). Let $P(n,d)$ be a set of $n$ pairs of chords of $C$ such that for each $\{c,c'\}\in P(n,...
- 1,677
3
votes
1
answer
143
views
Combinatorial Euclidean geometry problem
Let $\mathcal{S}^d_{\epsilon}$ be the collection of all sets $S:=\{\mathbf{x}_1, \mathbf{x}_2, \ldots \mathbf{x}_{d+1}\}$ of $d+1$ points in a $d$-dimensional Euclidean space such that, for a given ...
- 1,677
6
votes
1
answer
176
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Subsets of a ball/sphere with the largest sum of distances
$\newcommand\R{\mathbb R}\newcommand\S{\mathbb S}$Let $B_d$ and $S_{d-1}$ denote, respectively, the closed unit ball and the unit sphere in $\R^d$. Let us say that a finite subset $F$ of $B_d$ is ...
- 128k
2
votes
2
answers
379
views
Probabilistic combinatorial optimization problem on the distances between pairs of points in $[0,1]$
Let $S$ be a set of $n \gg 1$ points lying on the interval $[0,1]$. Given a point $p\in[0,1]$, let $S_p\subseteq S\times S$ be the set formed by all pairs of points $(x,y)$ with $x,y\in S$, such that ...
- 1,677
5
votes
0
answers
199
views
Existence of a honeycomb composed by nearly-hyperspherical $d$-dimensional cells having the same shape and size
Let $\mathcal{H}$ the class of all honeycombs composed by $d$-dimensional cells $C$ having all the same shape and size in a $d$-dimensional space $\mathcal{S}$.
Let $s(C)$ and $\ell(C)$ be ...
- 1,677
4
votes
1
answer
363
views
Trade-off between hypervolume and diameter of $d$-dimensional shapes having a hypercubic smallest bounding box
Given any $d$-dimensional shape $X$, let $V(X)$ be its $d$-dimensional volume, and let $\ell(X)$ be the length of the longest line segment connecting two points of $X$.
Let $\mathcal{S}_C$ be the set ...
- 1,677
2
votes
3
answers
290
views
Geometric probabilistic problem on triangles on a plane
We are given a triangle $T$ on a plane $P$, with sidelengths $a$, $b$ and $c$, where $c \ge b \ge a > 0$. A straight line $L$ on $P$ is selected uniformly at random from the set of all the ...
- 1,677
4
votes
0
answers
246
views
Distance properties of the permutations of a set of points in a Euclidean space
We are given a set of $n$ distinct points $S=\{\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n\}$ in a Euclidean space $\mathbb{R}^d$, where the distance between two points $\mathbf{x}_i,\mathbf{x}_j\...
- 1,677
2
votes
1
answer
112
views
Find the largest subset of all binary arrays of length $n$ with $r$ ones which have pairwise distance greater than $m$
Let $\Omega = \left\lbrace x : |x| = r \, \text{for} \, x \in \mathbb{Z}_2^n \right\rbrace$ for $r \in \mathbb{N}$. We want to find the the biggest subset of $\Omega$, $\Gamma = \left\lbrace x \in \...
5
votes
1
answer
415
views
Spreading $n$ points in $\{0,1\}^n$ as far as possible
Given a positive integer $n$, the Hamming distance $d^H_n(x,y)$ of $x,y\in \{0,1\}^n$ is defined by $$d^H_n(x,y) = |\{k\in\{0,\ldots,n-1\}: x(k)\neq y(k)\}|.$$
We say that a positive integer $s$ is $...
- 48.9k
12
votes
1
answer
214
views
The angles subtended in a TSP tour
If I sample a large number of uniform points in the unit square and take a traveling salesman tour of them, is there anything at all that can be said/is known about the distribution of angles at each ...
- 4,049
2
votes
0
answers
46
views
What is known about the relation of the multiple salesman problem and the travelling saelsman problem?
The travelling salesman problem (TSP) is well known (https://en.wikipedia.org/wiki/Travelling_salesman_problem). Let us look at the Euclidean TSP throughout this question.
There is a generalization ...
- 61
4
votes
3
answers
837
views
What is known about worst-case point sets for the travelling salesman problem?
The travelling salesman problem (TSP) is well-known, see e.g. https://en.wikipedia.org/wiki/Travelling_salesman_problem.
Let us consider the Euclidean version of the TSP within the unit square.
This ...
- 61
4
votes
1
answer
242
views
Closest point to a dual lattice point (in particular for root lattices!)
Given a lattice $\Lambda\subset \mathbb{R}^n$ and a point $p\in\mathbb{R}^n$ outside the lattice, then I known it is a hard question to determine the set $S\subset \Lambda$ of all lattice points with ...
- 1,846
1
vote
0
answers
96
views
Shortest paths stepping on rational points of height $h$
Q. Do shortest paths walking between rational points of height $h$
ever properly cross themselves?
Explaining this question takes a bit of definitional exposition.
First, I copy definitions from ...
- 151k
4
votes
0
answers
98
views
Nice minimal embeddings of large finite groups into compact Riemannian manifolds
The initial motivation for this question is a very practical problem: I need to find the absolute minimum of a function on a very large symmetric group $\Sigma_N$ (with $N$ 10000 or more). So if ...
- 18.4k
2
votes
2
answers
411
views
Upper bounds on the worst-case traveling salesman tours in the unit square
The paper [1] proves that, if we place $N$ points in the unit square, then the length $\ell$ of the euclidean TSP tour of those points must satisfy $$\ell \leq \sqrt{2N} + 7/4~~.$$ I'm wondering, can ...
4
votes
0
answers
163
views
Convexified threshold of a function
Upd. The question in a nutshell: find convex set on plane which is «closest» to a given non-convex set.
It is given integrable function $0\leq f(x,y)\leq 1$ with bounded support: $f(x,y)=0$ when $x^2+...
7
votes
3
answers
805
views
Wrapping a convex polyhedron with string
This is a meta-question, rather than a specific mathematical question.
I am seeking a mathematical definition that captures the following physical idea.
Suppose you have a convex polyhedron $P \...
- 151k
4
votes
1
answer
1k
views
Algorithm for the shortest path through all the points of a 2D cloud
I have an array of points with their coordinates X and Y. Each point represents a bus stop.
I need to sort the points in a sequence by giving them sequence numbers, so that the path from the first to ...
- 61
3
votes
2
answers
384
views
High dimensional Steiner tree
Given n affinely independent points in n-1 dimensional Euclidean space, how is the minimum Steiner tree constructed? Or assuming that the topology of the Steiner tree is given, is there an easy way ...
- 31