Mathematics

Questions tagged [birkhoff-polytopes]

The Birkhoff polytope is a convex polytope whose points are doubly stochastic matrices and whose vertices are permutation matrices.

38 questions
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Does Birkhoff - von Neumann imply any of the fundamental theorems in combinatorics?

I recently had the occasion to think about Hall's Marriage Theorem for the first time since my undergraduate combinatorics class more than a decade ago. Reading the wikipedia article linked above, I was interested to see that it is regarded as…
Pete L. Clark
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How do I generate doubly-stochastic matrices uniform randomly?

A doubly-stochastic matrix is an $n \times n$ matrix $P$ such that $$ \sum_{i=1}^n p_{ij} = \sum_{j=1}^n p_{ij} = 1 $$ where $p_{ij}\ge 0$ . Can someone please suggest an algorithm for generating these matrices uniform randomly?
Henry B.
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What's the algorithm of finding the convex combination of permutation matrices for a doubly stochastic matrix?

According to Birkhoff, $n$ -by- $n$ stochastic matrices form a convex polytope whose extreme points are precisely the permutation matrices. It implies that any doubly stochastic matrix can be written as a convex combination of finitely many…
xzhu
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Proof that the set of doubly-stochastic matrices forms a convex polytope?

Does the set of all doubly-stochastic matrices form a convex polytope? In general, I wonder how the proofs of convexity and geometry can be established for sets of matrices of this kind? Anything to do with the birkhoff's theorem!? Am not sure, if I…
qlinck
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Projection onto Birkhoff Polytope

Suppose we would like to compute the Euclidean projection of an arbitrary matrix $A$ onto the Birkhoff polytope, the set of doubly-stochastic matrices. Under some conditions on $A$, Sinkhorn's algorithm returns two diagonal matrices $D_1,D_2$ such…
cong
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$n$-sphere enclosing the Birkhoff polytope

I am not a mathematician by training, so please feel free to correct my logic or descriptions when necessary: Let $P$ denote a $\textit{permutation matrix}$ : $$ \begin{equation} P := \{X \in \{0,1\}^{n\times n} : X \mathbf{1}_n =…
Tolga Birdal
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How wide is the Birkhoff Polytope?

Now also posted on Math Overflow. Define the width of a polytope $P \subset \mathbb R^d$ as the minimum length of the interval $\{v \cdot p:p \in P\}$ for $v$ in the unit sphere. In other words the width is the smallest number $W$ such that you can…
Daron
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Birkhoff representation of a stochastic matrix

From the Birkhoff theorem , it is known that every doubly stochastic matrix can be written as a convex combination of permutation matrices, although this representation might not be unique. Assume that a stochastic matrix is given. How can I find a…
Saeid Haghighatshoar
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5
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The permutation matrices are the doubly stochastic matrices with the highest Frobenius norm

In a 2013 talk , Alexandre d'Aspremont did claim the following: Among all doubly stochastic matrices, the rotations, hence, the permutation matrices, have the highest Frobenius norm I had never encountered this result. Searching for it on…
Rodrigo de Azevedo
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What are the facets of the Birkhoff Polytope when $n=2$?

I've read in several sources that the number of facets of the Birkhoff polytope $\mathcal{B}(n)$ is $n^2$. Is this supposed to hold when $n=2$? Since $\mathcal{B}(2)$ has dimension $1$, the facets would be the two $0$-dimensional vertices, which are…
M47145
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Birkhoff-von Neumann Theorem

I am reading from Linear Algebra in Action by Dym and am working through the proof of the Birkhoff-von Neumann Theorem. For the sake of clarity, ill write up everything until the point where I get lost. Theorem: Let $P \in \mathbb{R}^{n\times n}$ be…
Alex Sampson
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Is the Birkhoff–von Neumann theorem true for infinite matrices?

The Birkhoff–von Neumann theorem states that every $n \times n$ doubly stochastic matrix is a convex combination of permutation matrices. Is this true for $\mathbb{N} \times \mathbb{N}$ matrices as well? If so, can you provide a reference?
Yoav Rodeh
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Can Sinkhorn's algorithm reach any point on the Birkhoff polytope?

It is well known that Sinkhorn's algorithm converges to a doubly stochastic matrix given a non-negative square input matrix. Knowing that Sinkhorn's algorithm always produces a DSM, I am interested in the opposite: Can every possible DSM be produced…
dopexxx
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Finding all integer solutions for a Linear Program over the Birkhoff Polytope

I have the situation where the Birkhoff Polytope is the space of all valid solutions to a linear function I am interested in maximizing. It is my understanding that, because the vertices of the Birkhoff polytope are the permutation matrices,…
Michael Keller
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Prove that a matrix can be written as a sum of permutation matrices

Given a square matrix $A$ of size $n$ whose entries are non-negative integers and where the sum of each column and row is equal to $k$ , prove that $A$ can be written as a sum of $k$ permutation matrices. First, it is obvious to see that a sum o $k$…
314canelas
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