Statements (49)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:Polygon
|
| gptkbp:alsoKnownAs |
gptkb:assignment_polytope
|
| gptkbp:application |
used in the study of Markov chains
used in the study of combinatorial optimization used in the study of convex geometry used in the study of graph theory used in the study of linear programming used in the study of matching problems used in the study of matrix theory used in the study of network flows used in the study of probability theory used in the study of quantum information theory used in the study of statistics used in the study of transportation problems |
| gptkbp:Birkhoff–von_Neumann_theorem |
every doubly stochastic matrix is a convex combination of permutation matrices
|
| gptkbp:category |
combinatorial optimization
matrix theory convex polytope |
| gptkbp:definedIn |
the convex hull of all n x n permutation matrices
|
| gptkbp:dimensionFor2x2 |
1
|
| gptkbp:dimensionFor3x3 |
4
|
| gptkbp:dimensions |
(n-1)^2 for n x n matrices
|
| gptkbp:faced |
correspond to partial permutation matrices
|
| gptkbp:facetCountFor2x2 |
4
|
| gptkbp:facetCountFor3x3 |
18
|
| gptkbp:facetCountFor4x4 |
72
|
| gptkbp:facetDescription |
x_{ij} ≥ 0, sum over rows and columns equals 1
facets correspond to the inequalities defining doubly stochastic matrices |
| https://www.w3.org/2000/01/rdf-schema#label |
Birkhoff polytope
|
| gptkbp:namedAfter |
gptkb:Garrett_Birkhoff
|
| gptkbp:property |
all points are doubly stochastic matrices
is a 0-1 polytope for n=2 is a subset of the n^2-dimensional real space is not a 0-1 polytope for n>2 vertices are exactly the permutation matrices |
| gptkbp:relatedTo |
gptkb:doubly_stochastic_matrix
gptkb:matching_polytope gptkb:transportation_polytope permutation matrix assignment problem doubly stochastic matrices |
| gptkbp:theory |
gptkb:Birkhoff–von_Neumann_theorem
|
| gptkbp:usedIn |
combinatorics
optimization linear programming |
| gptkbp:vertices |
permutation matrices
|
| gptkbp:bfsParent |
gptkb:George_D._Birkhoff
gptkb:George_David_Birkhoff |
| gptkbp:bfsLayer |
5
|