Birkhoff polytope

The Birkhoff polytope B_n, also called the assignment polytope, the polytope of doubly stochastic matrices, or the perfect matching polytope of the complete bipartite graph $K_{{n,n}}$ ,^[1] is the convex polytope in R^N (where N = n²) whose points are the doubly stochastic matrices, i.e., the n × n matrices whose entries are non-negative real numbers and whose rows and columns each add up to 1.

Properties

Vertices

The Birkhoff polytope has n! vertices, one for each permutation on n items.^[1] This follows from the Birkhoff–von Neumann theorem, which states that the extreme points 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,^[2] but equivalent results in the languages of projective configurations and of regular bipartite graph matchings, respectively, were shown much earlier in 1894 in Ernst Steinitz's thesis and in 1916 by Dénes Kőnig.^[3]

Edges

The edges of the Birkhoff polytope correspond to pairs of permutations differing by a cycle:

(\sigma ,\omega )

such that

\sigma ^{{-1}}\omega

is a cycle.

This implies that the graph of B_n is a Cayley graph of the symmetric group S_n. This also implies that the graph of B₃ is a complete graph K₆, and thus B₃ is a neighborly polytope.

Facets

The Birkhoff polytope lies within an (n² − 2n + 1)-dimensional affine subspace of the n²-dimensional space of all n × n matrices: this subspace is determined by the linear equality constraints that the sum of each row and of each column be one. Within this subspace, it is defined by n² linear inequalities, one for each coordinate of the matrix, specifying that the coordinate be non-negative. Therefore, it has exactly n² facets.^[1]

Symmetries

The Birkhoff polytope B_n is both vertex-transitive and facet-transitive (i.e. the dual polytope is vertex-transitive). It is not regular for n>2.

Volume

An outstanding problem is to find the volume of the Birkhoff polytopes. This has been done for n ≤ 10.^[4] It is known to be equal to the volume of a polytope associated with standard Young tableaux.^[5] A combinatorial formula for all n was given in 2007.^[6] The following asymptotic formula was found by Rodney Canfield and Brendan McKay:^[7]

{\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).

Ehrhart polynomial

Determining the Ehrhart polynomial of a polytope is harder than determining its volume, since the volume can easily be computed from the leading coefficient of the Ehrhart polynomial. The Ehrhart polynomial associated with the Birkhoff polytope is only known for small values, and it is only conjectured that all the coefficients of the Ehrhart polynomials are non-negative.

Generalizations

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 tables; they play an important role in Bayesian statistics.
The Birkhoff polytope is a special case of the matching polytope, defined as a convex hull of the perfect matchings 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.

References

1 2 3 Ziegler, Günter M. (2007) [2006], Lectures on Polytopes, Graduate Texts in Mathematics, 152 (7th printing of 1st ed.), New York: Springer, p. 20, ISBN 978-0-387-94365-7
↑ Birkhoff, Garrett (1946), "Tres observaciones sobre el algebra lineal [Three observations on linear algebra]", Univ. Nac. Tucumán. Revista A., 5: 147–151, MR 0020547 .
↑ Kőnig, Dénes (1916), "Gráfok és alkalmazásuk a determinánsok és a halmazok elméletére", Matematikai és Természettudományi Értesítő, 34: 104–119 .
↑ The Volumes of Birkhoff polytopes for n ≤ 10.
↑ Pak, Igor (2000), "Four questions on Birkhoff polytope", Annals of Combinatorics, 4: 83–90, doi:10.1007/PL00001277 .
↑ De Loera, Jesus A.; Liu, Fu; Yoshida, Ruriko (2007). "Formulas for the volumes of the polytope of doubly-stochastic matrices and its faces". arXiv:math.CO/0701866. .
↑ Canfield, E. Rodney; McKay, Brendan D. (2007). "The asymptotic volume of the Birkhoff polytope". arXiv:0705.2422. .

Additional reading

Matthias Beck and Dennis Pixton (2003), The Ehrhart polynomial of the Birkhoff polytope, Discrete and Computational Geometry, Vol. 30, pp. 623–637. The volume of B₉.

External links

Birkhoff polytope Web site by Dennis Pixton and Matthias Beck, with links to articles and volumes.

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