All Questions
Tagged with co.combinatorics packing-and-covering
16 questions with no upvoted or accepted answers
13
votes
0
answers
751
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$\epsilon$-nets with respect to the cut norm
The cut norm $||A||\_C$ of a real matrix $A = (a_{i,j}) \in \mathcal{R}^{n\times n}$ is the maximum over all $I \subseteq [n], J \subseteq [n]$ of the quantity $\left|\sum_{i \in I, j \in J}a_{i,j}\...
- 794
10
votes
0
answers
497
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Which finite sets could be packed into a square?
This question is inspired by an interesting visualization of the finite levels of von Neumann's hierarchy on Adam P. Goucher's blog, Complex Projective 4-Space.
The problem starts with a two-...
9
votes
0
answers
193
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Asymptotics of packing
Define $m(n,k,l)$ as the maximal size of a family $k$-element subsets of $[n]$ having the property that the intersection of every two sets is less than $l$.
As stated on wikipedia, in 1985 Rödl ...
- 393
6
votes
0
answers
1k
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How to pack 27 $a\times b\times c$ blocks into a cube of side $a+b+c$ with some kind of symmetry?
Recently I stumbled on the problem quoted here about a geometric proof of the AM-GM inequality $$(a_1+\cdots+a_n)^n\ge n^n a_1\cdots a_n$$ by packing $n^n$ rectangular $ n$-dimensional boxes of sides $...
- 13.4k
6
votes
0
answers
199
views
A polynomial counting some packings in $\mathbb Z/N\mathbb Z$
Given two integers $n$ and $N$ such that $N>{n+1\choose 2}$, we denote by $\alpha_n(N)$
the number of elements $(x_1,\dots,x_n)$ in $(\mathbb Z/N\mathbb Z)^n$ such that the
$2n$ elements $x_1,x_1+1,...
- 17.9k
5
votes
0
answers
313
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Trade-off between covering number, ball radius and diameter of $d$-dimensional shapes
Given any $d$-dimensional shape $X$ in the Euclidean space, let $\ell(X)$ be the length of the longest line segment connecting two points of $X$. How can we prove the following statement?
There exists ...
- 1,677
5
votes
0
answers
199
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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
0
answers
144
views
Approximation of a convex shape in the $d$-dimensional Euclidean space for $d\gg 1$
We are given a convex shape $C$ lying inside the hypercube $[0,1]^d$ in the $d$-dimensional Euclidean space. Let the volume of $C$ be $\tfrac12$ (I guess nothing changes for any other fixed constant ...
- 1,677
4
votes
0
answers
146
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Tiling squares with oblongs
An oblong is a rectangle whose width and length are consecutive integers: 1x2, 2x3, 3x4, etc. Does N exist such that it is possible to split the first N oblongs into 2 or more non-intersecting sets so ...
- 3,707
3
votes
0
answers
134
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Two questions on counterexamples to Borsuk's conjecture and ball-packings
In 1933 Karol Borsuk conjectured the following
Can every bounded subset $E$ of $\mathbb{R}^d$ be partitioned into $(d+1)$ sets, each of which has a smaller diameter than $E$?
Whilst new to this ...
- 31
2
votes
0
answers
48
views
Maximum coverage of an orthogonal polygon using $k$ rectangles
I have an orthogonal polygon (all edges are horizontal or vertical) which is convex (no holes in any row of column of the polygon).
I would like to cover as much as possible of this orthogonal polygon ...
- 21
2
votes
0
answers
39
views
Covering base sets $X$ with a subset family satisfying a "partial covering property"
Let $X$ be an $n$ element base set. Suppose I have a subset family $\mathscr{F} \subset 2^X$ satisfying the following property:
(*) For any subset $Y \subset X$, we can find an element $F \in \mathscr{...
- 384
2
votes
0
answers
131
views
Optimal way to group points in the plane into clusters
Consider a strictly decreasing sequence $d = (d_k)_{k\ge 1}$ of distances in $(0,1)$. Given a constant $C>2$, we say that $d$ has the $C$-grouping property if any finite non-empty subset $S$ (of ...
- 1,761
1
vote
0
answers
137
views
A comparison between packing and covering as classes of problems
We continue from Bounds for the Dispersal Problem in convex regions and Bounds for minimax facility location in a convex region
Let us consider the classes of problems:
Given a convex region $R$ and ...
- 5,979
1
vote
0
answers
124
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The smallest disk containing all cirular arcs
In a comment to my recent question about covering segments by a disk, Gerhard Paseman has suggested a generalisation: replacing the segments of the original $n$-gon by a simple closed (say, convex) ...
- 13.4k
0
votes
0
answers
189
views
Packing Icons Onto A screen
You are trying to pack icons onto a screen that is divided into n horizontal rows of uniformly varying size. The rows narrow by a fixed ratio as one goes up the screen from the bottom. Since the icons ...
