Standard pairs and group relaxations in integer programming

Abstract

The main result of this paper is a non-Buchberger algorithm for constructing initial ideals and Gröbner bases of toric ideals, based on the connections between toric ideals and integer programming. The tools used are those of standard pair decompositions of standard monomials of a toric initial ideal, localizations of such ideals at their associated primes and group relaxations of integer programs. We give an algorithm for constructing standard pair decompositions, provide degree bounds for certain elements in the reduced Gröbner bases of toric ideals, and derive bounds on the arithmetic degree of initial ideals of monomial curves. We also exhibit new results for the localizations of initial ideals arising from toric ideals of codimension two.