Questions tagged [polytope]

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Is the edge cover polytope integral on graphs with self-loops?

It is well known that the edge cover polytope is integral on simple graphs. I am wondering whether this also holds for graphs with self-loops. Here is a Linear Relaxation of the edge cover polytope, ...
10
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0answers
144 views

Is counting the total number of faces of a polytope $\#P$ hard?

Let $P$ be a polytope defined by $Ax = b, x \geq 0$. Question: What is the complexity of computing the total number of faces of $P$? I know counting vertices is $\# P$-complete, but this problem is ...
2
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1answer
151 views

Finding whether $n$ polytopes have nontrivial intersection from pairwise comparisons

I have a set of $n$ convex polytopes of the form $$\mathcal{L_i} = \{ \beta \mid C_i \beta \leq 0 \}$$ where $C$ is a matrix and $\beta$ is a vector. I know that for each pair of polytopes $$(\...
5
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1answer
153 views

Complexity of Finding Largest Set of Intersecting Convex Polytopes

I have a set of $n$ convex polytopes, and I wish to find the largest subset of those polytopes that shares at least one point in common. I think that this problem should be NP-hard, but I am ...
15
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1answer
656 views

Can one efficiently uniformly sample a neighbor of a vertex in the graph of a polytope?

I have a polytope $P$ defined by $\{ x : Ax \leq b, x \geq 0\}$ . Question: Given a vertex $v$ of $P$, is there a polynomial time algorithm to uniformly sample from the neighbors of $v$ in the graph ...
7
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1answer
288 views

Reaching the double exponential upper bound in Fourier-Motzkin elimination

One round of Fourier-Motzkin elimination may yield $n^2/4$ inequalities where $n$ is the original number of linear inequalities, whence an upper bound of $n^{2^d}/2^{2^{d+1}-2}$ for $d$ rounds of ...
0
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1answer
134 views

On polytope lattice points

Given a convex polytope let the width of the polytope be $d$ and the farthest euclidean distance between any points in the polytope be $e$. Denote $\mathcal P(a,c)$ to be the set of convex polytopes ...
9
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0answers
114 views

Complexity of checking the equivalence of constraints and generator descriptions of convex polyhedra

Consider the space $\mathbb{Q}^n$. A convex polyhedron is defined, equivalently, by a system of linear (in)equalities (with integer coefficients) or by a system of generators: vertices, and in case ...
4
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1answer
66 views

Reference needed for lower bound on number of guards in three-dimensional art gallery guarding

During my research (writing my master's thesis) I've stumbled across the book Art Gallery Theorems and Algorithms by O’Rourke. In chapter 10 (pdf available on the above site), section 10.2.2. he shows ...
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1answer
94 views

Properties of convex polytope of 0-1 matrices

Problem setting Consider a set $ S = \big\{ 1,2,\cdots,n \big\}$. Now consider $k$ equal-sized subsets $S_i \subset S$ s.t of size $\big|S_i\big|=n' \;\forall i$. Consider a $k\times k$ matrix $M$ ...
10
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0answers
132 views

Which convex polytopes have volumes of polynomial bit-length?

A convex polytope is described as an intersection of halfspaces, given as inequalities between linear combinations of variables with rational coefficients. The volume computation problem for convex ...
3
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0answers
106 views

Equivalence of weighted Minkowski sums

Given $n$ polytopes $P_1, \cdots, P_n$, each $P_i$ is given as the V-representation, i.e., a set of $m$ points as its set of vertices. Furthermore, consider a variant of the Minkowski sum (somehow ...
7
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1answer
348 views

Decide whether a point is a vertex of a polytope?

Inspired by the question, I would like to ask the following question: Input: A polytope specified by $\Theta=\{\vec{x}\mid A\vec{x}\leq b\}$, and its affine projection $f(\Theta)= \{(\vec{c}_1\cdot \...
2
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0answers
71 views

Check whether a point is a vertex of Minkowski sum of polytopes

Given $n$ polytopes $$\begin{align*} P_1&=\{(f^1_1(\vec{x}_1), \cdots, f^1_m(\vec{x}_1))\mid A_1\vec{x}_1\leq b_1\}\\ P_2&=\{(f^2_1(\vec{x}_2), \cdots, f^2_m(\vec{x}_2))\mid A_2\vec{x}_2\leq ...
14
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2answers
999 views

Checking equivalence of two polytopes

Consider a vector of variables $\vec{x}$, and a set of linear constraints specified by $A\vec{x}\leq b$. Furthermore, consider two polytopes $$\begin{align*} P_1&=\{(f_1(\vec{x}), \cdots, f_m(\...
-1
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1answer
283 views

Vertices of a polytope

Consider the polytope $P=\{(x_1,x_2,...,x_n)\in \mathbb{R}^n| \sum_{i=1}^n x_i=1; 0\leq a_i\leq x_i\leq b_i, i=1,...,n\}$ where $a_i$ and $b_i$ are constant lower and upper bounds for $x_i$. Is it ...
4
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1answer
185 views

When are all facets rank facets? (for independence system polyhedra)

Consider an independence system $(E,\mathcal{I})$, and the corresponding polytope: $P(E,\mathcal{I}):=\operatorname{conv.hull}\{ x^S ~|~S\in \mathcal{I}\}$ where $x^S \in \{0,1\}^E$ denotes the ...
8
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1answer
491 views

Is the feasible region of this SDP polyhedral?

We have a semidefinite program (SDP) whose feasible region contains only a finite number of rank-$1$ matrices. Can we conclude that the feasible region of this SDP is polyhedral? We believe this to ...
5
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0answers
146 views

Computing diameter of a 3D polyhedron

A polyhedron is given as a set of its vertex coordinates. Is it possible to find its diameter faster than $O(n^2)$? Or, maybe, some another common polyhedron representation would help fasten this?
2
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3answers
272 views

The number of integral points in a polytope

Suppose we define a polytope with $$ \mathbf{Ax} \leq \mathbf{b} $$ What is the best way to find/approximate the number of the integral points in the polytope? Update: how hard is the complexity ...
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0answers
133 views

An interesting class of colored graphs?

Let $G$ be a complete graph edge-colored with $k$ colors. We say that $G$ is Gallai-colored if no triangle is colored with three distinct colors. Fix a tuple of integers $c = (c_1,\ldots,c_k)$. We may ...
3
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1answer
192 views

What is known about this binary representation polytope?

Consider the following set. $S_n =\{ (x,y) ~|~x \in \mathbb{Z}_+~\wedge~ y \in \{0,1\}^n ~\wedge~x=\sum_{i=0}^{n-1} 2^i y_i \}$ $S_n$ is a collection of pairs $(x,y)$, where $x$ is an integer ...