A Variant of the Buchberger Algorithm for Integer Programming

In this paper we modify Buchberger's S-pair reduction algorithm for computing a Grobner basis of a toric ideal so as to apply it to an integer program (IP) in inequality form with fixed right-hand sides and fixed upper bounds on the variables. We formulate the algorithm in the original space and interpret the reduction steps geometrically. In fact, three variants of this algorithm are presented, and we give elementary proofs for their correctness. A relationship among these (exact) algorithms, iterative improvement heuristics, and the Kernighan--Lin procedure is established. Computational results are also presented.