Diophantine Approximation and Basis Reduction

Abstract

As mentioned in Chapter 1, combinatorial optimization problems can usually be formulated as linear programs with integrality constraints. The geometric notion reflecting the main issues in linear programming is convexity, and we have discussed the main algorithmic problems on convex sets in the previous chapters. It turns out that it is also useful to formulate integrality constraints in a geometric way. This leads us to “lattices of points”. Such lattices have been studied (mostly from a nonalgorithmic point of view) in the “geometry of numbers”; their main application has been the theory of simultaneous diophantine approximation, i. e., the problem of approximating a set of real numbers by rational numbers with a common small denominator. We offer an algorithmic study of lattices and diophantine approximation.