A geometric Buchberger algorithm for integer programming

Let IP{A,c} denote the family of integer programs of the form Min cx: Ax = b, x is an element of N-n obtained by varying the right-hand side vector tb but keeping A and c fixed. A test set for IP{A,c} is a set of vectors in Z(n) such that for each nonoptimal solution alpha to a program in this family, there is at least one element g in this set such that alpha - g has an improved cost value as compared to alpha. We describe a unique minimal test set for this family called the reduced Grobner basis of IP{A,c} . An algorithm for its construction is presented which we call a geometric Buchberger algorithm for integer programming and we show how an integer program may be solved using this test set. The reduced Grobner 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 is an element of Z((n-2)xn) of full row rank.