Basis Reduction Methods

where P is a polyhedron described by inequalities. The work of Lenstra in 1979 [1] answered one of the most challenging questions in the theory of integer programming by presenting an algorithm to solve (IP), whose running time is polynomial, when n, the dimension is fixed. This paper pioneered the use of lattice basis reduction in integer programming, and initiated an interest in polynomial results in integer programming under the ‘‘fixed dimension’’ assumption. Somewhat later Kannan developed an improved variant [2,3], which—to date—has the best theoretical complexity for integer programming feasibility. The goal of this survey is to review lattice-based methods to solve (IP), focusing on Lenstra’s and Kannan’s algorithms, which are by now considered ‘‘classical,’’ and the more recent reformulation methods of Aardal et al. [4], and Krishnamoorthy and Pataki [5].