A Geometric Buchberger Algorithm for Integer Programming

Let IP denote the family of integer programs of the form Min cx : Ax = b, x 2 N n obtained by varying the right hand side vector b but keeping A and c xed. A test set for IP is a set of vectors in Z n such that for each non-optimal solution to a program in this family, there is at least one element g in this set such that ? g has an improved cost value as compared to. We describe a unique minimal test set for this family called the reduced Grr obner basis of IP. An algorithm for its construction is presented which we call a Geometric Buchberger Algorithm for integer programming. We show how an integer program may be solved using this test set and examine some geometric properties of elements in the set. The reduced Grr obner basis is then compared with some other known test sets from the literature. We also indicate an easy procedure to construct test sets with respect to all cost functions for a matrix A 2 Z (n?2)n of full row rank.