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3 votes
0 answers
50 views

Modular counting of integral points under sparse non-negativity

Given a polyhedron $$Ax\geq b$$ where every entry of $A,b$ are non-negative and $A\in\{0,1\}^{m\times n}$ and there are $O(1)$ (say $\leq8$) non-negative entries per row of $A$ is it possible to ...
Turbo's user avatar
  • 13.9k
1 vote
0 answers
38 views

Structural properties of polytopes for mainstream integer or linear programs

Are there any papers/textbooks/monographs that describe distinguishing properties of the polytopes that arise when solving the linear relaxation of well-known integer programs? For example, it is ...
Tom Solberg's user avatar
  • 4,049
0 votes
0 answers
27 views

How do you refer to the feasible set of solutions to a mixed-integer program?

I frequently want to refer to the feasible set of solutions to a mixed integer programming instance. Is there a name for a subset of $\mathbb{R}^n\times\{0,1\}^m$ of the form $\{(x,a)| Ax + Ba\leq b\}$...
bucket's user avatar
  • 85
1 vote
0 answers
127 views

Mixed integer formulation of union of polytopes?

Given $t$ different unbounded polyhedra $P_1:A^{(1)}x^{(1)}\leq b^{(1)},\dots,P_t:A^{(t)}x^{(t)}\leq b^{(t)}$ we are looking for the representation of $\bigcup_{i=1}^tP_i$ (not their convex hull) with ...
VS.'s user avatar
  • 1,836
1 vote
0 answers
74 views

How to minimize n-polytope's bounding box with linear transformation?

I am working on an exact algorithm for integer linear programming for my master's thesis: $Ax\leq b, x \in \mathbb{Z}^n$ $cx\rightarrow min$ For my idea to work out, I need a guarantee that n-...
Иван Шумилов's user avatar
-1 votes
2 answers
114 views

On OR condition in Linear Programming with exponentially many constraints [closed]

Suppose we have two linear programs $Ax\leq b$ and $Bx\leq c$ is there a way to combine them into one program of possibly a larger dimension $Cy\leq d$ such that projection of vectors $y$ into a ...
Turbo's user avatar
  • 13.9k
0 votes
0 answers
369 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 \...
rasul's user avatar
  • 136
2 votes
1 answer
340 views

Bit complexity of Barvinok's algorithm

I have seen many references which state Barvinok's algorithm has polynomial time complexity for counting integer points of polytopes in fixed dimension. What exactly is this arithmetic complexity? ...
Turbo's user avatar
  • 13.9k
4 votes
1 answer
150 views

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 ...
Turbo's user avatar
  • 13.9k
0 votes
2 answers
318 views

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 $X$ ...
Alex's user avatar
  • 501
5 votes
0 answers
129 views

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 ...
Alex's user avatar
  • 501
2 votes
0 answers
163 views

existence of lattice point in polytope

This question was probably asked before but here goes. I have a convex polytope given by $Ax\leq b$ for a specific integer matrix $A$ and integer vector $b$. I need a simple method/result on how to ...
Alex's user avatar
  • 501
1 vote
1 answer
277 views

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\...
Shake Baby's user avatar
  • 1,638