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| The '''Birkhoff polytope''' ''B''<sub>''n''</sub>, also called the '''assignment polytope''', the '''polytope of doubly stochastic matrices''', or the '''perfect matching polytope''' of the [[complete bipartite graph]] <math>K_{n,n}</math>,<ref name="z">{{citation |last=Ziegler |first=Günter M. |authorlink=Günter M. Ziegler |title=Lectures on Polytopes |edition=7th printing of 1st |series=Graduate Texts in Mathematics |volume=152 |origyear=2006 |year=2007 |publisher=Springer |location=New York |isbn=978-0-387-94365-7 |page=20}}</ref> is the [[convex polytope]] in '''R'''<sup>''N''</sup> (where ''N'' = ''n''²) whose points are the [[doubly stochastic matrix|doubly stochastic matrices]], i.e., the {{nowrap|''n'' × ''n''}} matrices whose entries are non-negative real numbers and whose rows and columns each add up to 1. | | The writer is recognized by the title of Figures Wunder. She is a librarian but she's always wanted her personal company. For a while I've been in South Dakota and my parents live nearby. One of the extremely very best issues in the world for me is to do aerobics and I've been performing it for fairly a while.<br><br>Feel free to surf to my webpage - [http://www.youporn-nederlandse.com/user/CMidgett std testing at home] |
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| == Properties ==
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| ===Vertices===
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| The Birkhoff polytope has ''n''! vertices, one for each permutation on ''n'' items.<ref name="z"/> This follows from the [[Birkhoff–von Neumann theorem]], which states that the [[extreme point]]s of the Birkhoff polytope are the [[permutation matrices]], and therefore that any doubly stochastic matrix may be represented as a convex combination of permutation matrices; this was stated in a 1946 paper by [[Garrett Birkhoff]],<ref>{{citation | last = Birkhoff | first = Garrett | authorlink=Garrett Birkhoff |title = Tres observaciones sobre el algebra lineal [Three observations on linear algebra] | journal = Univ. Nac. Tucumán. Revista A. | volume = 5 | year = 1946 | pages = 147–151 | mr = 0020547 }}.</ref> but equivalent results in the languages of [[projective configuration]]s and of [[regular graph|regular]] [[bipartite graph]] [[Matching (graph theory)|matching]]s, respectively, were shown much earlier in 1894 in [[Ernst Steinitz]]'s thesis and in 1916 by [[Dénes Kőnig]].<ref>{{citation | last = Kőnig | first = Dénes | authorlink = Dénes Kőnig | title = Gráfok és alkalmazásuk a determinánsok és a halmazok elméletére | journal = Matematikai és Természettudományi Értesítő | volume = 34 | year = 1916 | pages = 104–119}}.</ref>
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| ===Edges===
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| The edges of the Birkhoff polytope correspond to pairs of permutations differing by a cycle:
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| : <math>(\sigma,\omega)</math> such that <math>\sigma^{-1}\omega</math> is a cycle.
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| This implies that the [[Graph (mathematics)|graph]] of ''B''<sub>''n''</sub> is a [[Cayley graph]] of the symmetric group ''S''<sub>''n''</sub>. This also implies that the graph of ''B''<sub>''3''</sub> is a [[complete graph]] ''K''<sub>''6''</sub>, and thus ''B''<sub>''3''</sub> is a [[neighborly polytope]].
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| ===Facets===
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| The Birkhoff polytope lies within an {{nowrap|(''n''<sup>2</sup> − 2''n'' + 1)-}}dimensional [[affine subspace]] of the ''n''<sup>2</sup>-dimensional space of all {{nowrap|''n'' × ''n''}} matrices: this subspace is determined by the linear equality constraints
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| that the sum of each row and of each column be one. Within this subspace, it is defined by ''n''<sup>2</sup> [[linear inequality|linear inequalities]], one for each coordinate of the matrix, specifying that the coordinate be non-negative.
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| Therefore, it has exactly ''n''<sup>2</sup> [[facet (geometry)|facets]].<ref name="z"/>
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| === Symmetries ===
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| The Birkhoff polytope ''B''<sub>''n''</sub> is both [[vertex-transitive]] and [[Isohedral_figure|facet-transitive]] (i.e. the [[Dual polyhedron|dual polytope]] is vertex-transitive). It is not [[regular polytope|regular]] for ''n>2''.
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| ===Volume===
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| An outstanding problem is to find the volume of the Birkhoff polytopes. This has been done for ''n'' ≤ 10.<ref>The [http://www.math.binghamton.edu/dennis/Birkhoff/volumes.html Volumes of Birkhoff polytopes] for ''n'' ≤ 10.</ref> It is known to be equal to the volume of a polytope associated with standard [[Young tableau]]x.<ref>{{citation |authorlink=Igor Pak |first=Igor |last=Pak |title=Four questions on Birkhoff polytope |journal=Annals of Combinatorics |volume=4 |year=2000 |pages=83–90 |doi=10.1007/PL00001277}}.</ref> A combinatorial formula for all ''n'' was given in 2007.<ref>{{cite arxiv |title=Formulas for the volumes of the polytope of doubly-stochastic matrices and its faces |first1=Jesus A. |last1=De Loera |first2=Fu |last2=Liu |first3=Ruriko |last3=Yoshida |year=2007 |eprint=math.CO/0701866}}.</ref> The following [[asymptotic formula]] was found by
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| [[Rodney Canfield]] and [[Brendan McKay]]:<ref>{{cite arxiv |first1=E. Rodney |last1=Canfield |authorlink2=Brendan McKay |first2=Brendan D. |last2=McKay |title=The asymptotic volume of the Birkhoff polytope |year=2007 |eprint=0705.2422}}.</ref>
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| :<math>\mathop{\mathrm{vol}}(B_n) \, = \, \exp\left( - (n-1)^2\ln n + n^2 - (n - \frac{1}{2})\ln(2\pi) + \frac{1}{3} + o(1) \right) .</math>
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| == Generalizations ==
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| *The Birkhoff polytope is a special case of the [[transportation polytope]], a polytope of nonnegative rectangular matrices with given row and column sums. The integer points in these polytopes are called [[contingency table]]s; they play an important role in [[Bayesian statistics]].
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| *The Birkhoff polytope is a special case of the [[matching polytope]], defined as a convex hull of the [[perfect matching]]s in a finite graph. The description of facets in this generality was given by [[Jack Edmonds]] (1965), and is related to [[Edmonds's matching algorithm]].
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| ==See also==
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| * [[Permutohedron]]
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| ==References==
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| {{reflist}}
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| ==Additional reading==
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| * Matthias Beck and Dennis Pixton (2003), The Ehrhart polynomial of the Birkhoff polytope, <cite>[[Discrete and Computational Geometry]]</cite>, Vol. 30, pp. 623–637. The volume of ''B''<sub>9</sub>.
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| ==External links==
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| * [http://www.math.binghamton.edu/dennis/Birkhoff/ Birkhoff polytope] Web site by Dennis Pixton and Matthias Beck, with links to articles and volumes.
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| [[Category:Polytopes]]
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| [[Category:Matrices]]
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The writer is recognized by the title of Figures Wunder. She is a librarian but she's always wanted her personal company. For a while I've been in South Dakota and my parents live nearby. One of the extremely very best issues in the world for me is to do aerobics and I've been performing it for fairly a while.
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