149 research outputs found
Computing parametric rational generating functions with a primal Barvinok algorithm
Computations with Barvinok's short rational generating functions are
traditionally being performed in the dual space, to avoid the combinatorial
complexity of inclusion--exclusion formulas for the intersecting proper faces
of cones. We prove that, on the level of indicator functions of polyhedra,
there is no need for using inclusion--exclusion formulas to account for
boundary effects: All linear identities in the space of indicator functions can
be purely expressed using half-open variants of the full-dimensional polyhedra
in the identity. This gives rise to a practically efficient, parametric
Barvinok algorithm in the primal space.Comment: 16 pages, 1 figure; v2: Minor corrections, new example and summary of
algorithm; submitted to journa
Challenging computations of Hilbert bases of cones associated with algebraic statistics
In this paper we present two independent computational proofs that the monoid
derived from contingency tables is normal, completing the
classification by Hibi and Ohsugi. We show that Vlach's vector disproving
normality for the monoid derived from contingency tables is
the unique minimal such vector up to symmetry. Finally, we compute the full
Hilbert basis of the cone associated with the non-normal monoid of the
semi-graphoid for . The computations are based on extensions of the
packages LattE-4ti2 and Normaliz.Comment: 10 page
A Generating Function for all Semi-Magic Squares and the Volume of the Birkhoff Polytope
We present a multivariate generating function for all n x n nonnegative
integral matrices with all row and column sums equal to a positive integer t,
the so called semi-magic squares. As a consequence we obtain formulas for all
coefficients of the Ehrhart polynomial of the polytope B_n of n x n
doubly-stochastic matrices, also known as the Birkhoff polytope. In particular
we derive formulas for the volumes of B_n and any of its faces.Comment: 24 pages, 1 figure. To appear in Journal of Algebraic Combinatoric
Integer Carath\'eodory results with bounded multiplicity
The integer Carath\'eodory rank of a pointed rational cone is the
smallest number such that every integer vector contained in is an
integral non-negative combination of at most Hilbert basis elements. We
investigate the integer Carath\'eodory rank of simplicial cones with respect to
their multiplicity, i.e., the determinant of the integral generators of the
cone. One of the main results states that simplicial cones with multiplicity
bounded by five have the integral Carath\'eodory property, that is, the integer
Carath\'eodory rank equals the dimension. Furthermore, we present a novel upper
bound on the integer Carath\'eodory rank which depends on the dimension and the
multiplicity. This bound improves upon the best known upper bound on the
integer Carath\'eodory rank if the dimension exceeds the multiplicity. At last,
we present special cones which have the integral Carath\'eodory property such
as certain dual cones of Gorenstein cones
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