We show that the complexity of the Markov bases of multidimensional tables stabilizes eventually if a single table dimension is allowed to vary. In particular, if this table dimension is beyond a computable bound, the Markov bases consist of elements of Markov bases of smaller tables. We give an explicit formula for this bound in terms of Graver bases. We also compute these Markov and Graver co...

It has been shown in a number of recent papers that Graver bases methods enable to solve linear and nonlinear integer programming problems in variable dimension in polynomial time, resulting in a variety of applications in operations research and statistics. In this article we continue this line of investigation and show that Graver bases also enable tominimize quadratic and higher degree polyn...

let $l$ be a lattice in $zz^n$ of dimension $m$. we prove that there exist integer constants $d$ and $m$ which are basis-independent such that the total degree of any graver element of $l$ is not greater than $m(n-m+1)md$. the case $m=1$ occurs precisely when $l$ is saturated, and in this case the bound is a reformulation of a well-known bound given by several authors. as a corollary, we show t...

Many challenging Graver bases computations, like for multi-way tables in statistics, have a highly symmetric problem structure that is not exploited so far computationally. In this paper we present a Graver basis algorithm for sublattices of Zn that exploits existing symmetry.

A universal Gröbner basis of an ideal is the union of all its reduced Gröbner bases. It is contained in the Graver basis, the set of all primitive elements. Obtaining an explicit description of either of these sets, or even a sharp degree bound for their elements, is a nontrivial task. In their ’95 paper, Graham, Diaconis and Sturmfels give a nice combinatorial description of the Graver basis f...

We present a new algebraic algorithmic scheme to solve convex integer maximization problems of the following form, where c is a convex function on R and w1x, . . . , wdx are linear forms on R, max {c(w1x, . . . , wdx) : Ax = b, x ∈ N} . This method works for arbitrary input data A, b, d, w1, . . . , wd, c. Moreover, for fixed d and several important classes of programs in variable dimension, we...

Linear codes over any finite field can be associated to binomial ideals. Focussing on two specific instances, called the code ideal and the generalized code ideal, we present how these binomial ideals can be computed from toric ideals by substituting some variables. Drawing on this result we further show how their Graver bases as well as their universal Gröbner bases can be computed.