All Questions
Tagged with integer-programming lattices
15 questions
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Existence of some lattice path connecting all given lattice paths
My daily work concerns analysis on metric spaces and some time ago it turned out that the problem I am dealing with boils down to a certain combinatorial problem. I've checked a lot of examples and it ...
- 137
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221
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Modular inverse computation - avoiding Euclidean algorithm
Modular inverse is known to be computable by Extended Euclidean algorithm which is the reaping the rewards of computing the GCD of two numbers or proving two numbers are coprime.
If we already know ...
- 13.9k
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96
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Smallest integer lattice point by box measure in a polytope?
Given an integer lattice $\mathcal L\subseteq\mathbb Z^n$ represented by basis $\mathcal B$ and an integer linear program $Ax\leq b$ where $x\in\mathbb Z^n$ is unknown with $A\in\mathbb Z^{m\times n}$ ...
- 13.9k
4
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150
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On necessary condition for no integer points in polytope
For a convex polytope $\mathcal K$ in $\Bbb R^n$ presented by $O(n^c)$ linear inequalities is it true that for $|\mathcal K\cap \Bbb Z^n|=0$ it is necessary that at least one axis of John's ellipsoid ...
- 13.9k
2
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Gap in Successive minima on lattice spanned by rational and integer combination of integer vectors
We are given a rank $r$ matrix $B\in\Bbb Z^{k\times n}$ where $0\leq r\leq k\leq n$ holds.
We have
$$\mathcal L_\Bbb Z=\{uB\in\Bbb Z^n:u\in\Bbb Z^k\}\subseteq\mathcal L_\Bbb Q=\{uB\in\Bbb Z^n:u\in\...
- 13.9k
2
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2
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105
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Integer points spanned by real, rational and integer combination of integer vectors
We are given a rank $r$ matrix $B\in\Bbb Z^{k\times n}$ where $0\leq r\leq k\leq n$ holds.
We have $\mathcal L_\Bbb Z\subseteq \mathcal L_\Bbb Q\subseteq\mathcal L_\Bbb R$ where
$$\mathcal L_\Bbb Z=\...
- 13.9k
0
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318
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Lattice question
Consider a lattice $\mathcal{L} = \mathbb{Z}v_1 \oplus \ldots \oplus \mathbb{Z}v_l$ in $\mathbb{R}^n$ and let $S_0$ be the set of edges of the fundamental unit of $\mathcal{L}$. We call a region $X$ ...
- 501
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129
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Lattice paths in polytopes
Let $P$ be a polytope in $\mathbb{R}^n$. Let $A_ix = b_i$ be the defining equations of its codimension $1$ faces. Is there an algorithm or some kind of criterion to decide if the lattice points inside ...
- 501
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On design of a (preferrably unimodular) matrix
Assume each entry is in $\Bbb Z$.
Say we want to solve $Ax=b$ where known $A$ is $n\times n$, unknown $x$ is $n\times1$ and $b$ is $n\times1$.
The absolute value of minors of augmented matrix $[A|b]$...
- 13.9k
3
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Lenstra's integer programming algorithm: Finding a lattice point “near the center”
I have already posted this question on the mathematics forum, but I suspect the question needs more detailed knowledge than most users have; please excuse the duplicate post. Any help is greatly ...
- 31
1
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2
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753
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basis of the lattice generated by the integer points inside a subspace of R^L
Consider $K$ linearly independent vectors $\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{a}_K \in \mathbb{Z}^L$, where $1 \leq K<L $. Hence, the span of $\lbrace\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{...
- 51
5
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635
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Area of a lattice polygon in terms of its width
Let $M$ be a lattice polygon on a plane (i.e. its vertices are integer points $(i,j)\in\mathbb Z^2$).
Let us define lattice width in a direction $v=(m,n)\in\mathbb Z^2$ as $w_v(M)=\max\limits_{x,y\in ...
- 5,063
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Representing integer points inside a polytope using a unit hypercube
Let $D\subset\mathbb{Z}\times\mathbb{Z}$ such that $D$ is formed by the points bounded by a convex polytope. Let $f:D\to\mathbb{R}$ defined by $f(x,y)=ax+by+c$, where $a,b,c\in\mathbb{R}$ and $g:D\to\...
- 1,638
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Domination in Nice Lattices
Let an integer vector be nice when it has only two nonzero components, which sum to zero. So (0, 0, 3, 0, -3) and (-1, 0, 1, 0, 0) are examples of nice vectors in $n=5$ dimensions.
Call a lattice ...
- 1,288
10
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Condition for existence of certain lattice points on polytopes
Let $a_1,\cdots, a_n$ be integers such that $a_i\geq 2$ for all $i$ and $k>0$ another integer.
I am interested in whether there exist integers $x_1,\cdots, x_n$ with $0<x_i<a_i$ satisfying:
...
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