On Lattice Methods in Integer Optimization

Abstract

Integer optimization is a powerful modeling tool both for problems of practical and more abstract origin. Since the 1970s we have seen huge progress in the size of problem instances that can be tackled. This progress is mostly due to the many results in polyhedral combinatorics and to algorithms and implementations related to the polyhedral results. In the theory of integer optimization we have also seen exciting results related to the algebraic structure of the set of integer points in polyhedra together with algorithms that exploit them. This thesis presents results that make a step in the direction of merging the approach of polyhedral combinatorics with a reformulation technique built on lattices, an algebraic concept generalizing the structure of the integer points.