Computing generating sets of lattice ideals and Markov bases of lattices

Abstract

In this article, we present a new algorithm for computing generating sets and Gröbner bases of lattice ideals. In contrast to other existing methods, our algorithm starts computing in projected subspaces and then iteratively lifts the results back into higher dimensions, by using a completion procedure, until the original dimension is reached. We give a completely geometric presentation of our Projectand-Lift algorithm and describe also the two other existing main algorithms in this geometric framework. We then give more details on an efficient implementation of this algorithm, in particular on critical-pair criteria specific to lattice ideal computations. Finally, we conclude the paper with a computational comparison of our implementation of the Project-and-Lift algorithm in 4ti2 with algorithms for lattice ideal computations implemented in CoCoA and Singular. Our algorithm outperforms the other algorithms in every single instance we have tried.Recently, there has been renewed interest in toric ideal computations because of applications in algebraic statistics. Here, we are interested in Markov bases, which are used in a Monte-Carlo Markov-Chain (MCMC) process to test validity of statistical models via sampling. Diaconis and Sturmfels (Diaconis and Sturmfels,where A σ and Aσ are the sub-matrices of A whose columns are indexed by σ andσ respectively, and A σ Z := {A σ z : z ∈ Z |σ| }. In the special case where σ = ∅, we set A σ Z := {0}, and the problem IP ∅ A,c,b simplifies to IP A,c,b := min{cz : Az = b, z ∈ N n }. Note that group relaxations and extended group relaxations of IP A,c,b are also of the form IP σ A,c,b for some cost vectorc ∈ Q |σ| (Hosten and Thomas, 2002). Without loss of generality, we assume that c is generic meaning that IP σ A,c,b has a unique optimal solution for every feasible b ∈ Z d . We can always easily perturb a given c so that it is generic.if T contains an improving direction t for every non-optimal feasible solution z ∈ N |σ| of IP σ A,c,b ; that is, z − t is also feasible and c(z − t) < cz. Clearly, z − t being feasible implies that t is an element of the latticeMoreover, a set T ⊆ L σ A is called a test set for IP σ A,c the class of integer programs IP σ A,c,b for all b ∈ Z d , if T is a test set for every integer program in IP σ A,c . Graver showed that there exist finite sets T that are test sets for IP A,c (σ = ∅). In fact, his sets also constitute finite test sets for IP σ A,c for arbitrary

Toric fiber products

“…Computing the minimal generators of the ideal K d = ker(φ d ) for various values of d is a benchmark problem for algorithms for computing Gröbner bases of toric ideals [10,12]. These ideals are extremely complicated and while there are explicit generating sets known for some special values of d [2], there is, at present, no uniform description of the generating sets of these ideals.…”

A saturation algorithm for homogeneous binomial ideals

“…We compare our algorithm with the Sturmfels' algorithm [3] and Project and Lift [2], the best algorithm known to date to compute toric ideals. As expected, the table shows that our algorithm performs much better than the Sturmfels' algorithm, as our algorithm is specifically designed for binomial ideals.…”

“…As expected, the table shows that our algorithm performs much better than the Sturmfels' algorithm, as our algorithm is specifically designed for binomial ideals. [2], without optimizations reported in the subsequent pages. Similar optimizations are applicable in our algorithm and it too is implemented without the same in these experiments.…”

“…The typical results are presented in the table given below. For a definitive comparison we intend to implement our algorithm with all possible optimizations and compare with 4ti2 [2], which is the optimal implementation of their algorithm.…”

Binomial fibers and indispensable binomials

“…Up to now, it has mainly been addressed for toric and lattice ideals, see [3,4,5,11,12,22] among others. In Section 1, we consider this problem in the case of binomial ideals.…”