Computing the integer programming gap

TL;DR: In this paper, an algorithm for the sequential sampling of entries in multiway contingency tables with given constraints is described, where sampling values at each step are related to properties of the associated toric ideal using computational commutative algebra.

TL;DR: In this paper, the problem of computing bounds for cell entries is revisited and a connection between the ideas on bounds and the existence of maximum likelihood estimates is made rigorous through the underlying mathematics of the same geometric/algebraic framework.

TL;DR: In this article, it was shown that there is an algorithm that solves this problem in polynomial time if $p$ and $n$ are fixed, where n is the affine dimension of the polyhedron.

TL;DR: In this paper, the authors characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characterization of such sets is also given.

TL;DR: Grobner basics The state polytope Variation of term orders Toric ideals Enumeration, sampling and integer programming Primitive partition identities Universal Grobner bases Regular triangulations The second hypersimplex $\mathcal A$-graded algebras Canonical subalgebra bases Generators, Betti numbers and localizations Toric varieties in algebraic geometry as mentioned in this paper.

TL;DR: The Berlekamp-Massey-Sakata Decoding Algorithm is used for solving Polynomial Equations and for computations in Local Rings.

TL;DR: LattE , a computer package for lattice point enumeration which contains the first implementation of A. Barvinok’s algorithm, is described and it is proved that these kinds of symbolic–algebraic ideas surpass the traditional branch-and-bound enumeration and in some instances LattE is the only software capable of counting.

TL;DR: In this paper, it was shown that the Frobenius problem can be solved in polynomial time for integer semigroups and Hilbert bases of rational cones, provided certain parameters (the dimension and the number of generators) are specified.