An Implementation of the Generalized Basis Reduction Algorithm for Integer Programming

Abstract

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: H. W. Lenstra (1983) 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 L. Lovász and H. E. 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, with 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. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.