An Implementation of the Generalized Basis Reduction Algorithm for Integer Programming

In recent years many advances have been made in solution techniques for specially structured 0–1 integer programming problems. In contrast, very little progress has been made on solving general (mixed integer) problems. This, of course, is not true when viewed from the theoretical side: Lenstra (1981) made a major breakthrough, obtaining a polynomial-time algorithm when the number of integer variables is fixed. We discuss a practical implementation of a Lenstra-like algorithm, based on the generalized basis reduction method of Lovasz and Scarf (1988). This method allows us to avoid the ellipsoidal approximations required in Lenstra’s algorithm. We report on the solution of a number of small (but difficult) examples, up to 100 integer variables. Our computer code uses the linear programming optimizer CPlex as a subroutine to solve the linear programming problems that arise.