It is common knowledge that the understanding of the combinatorial geometry of convex bodies has helped speed up algorithms in discrete optimization. For example, cutting planes and facet-description of polyhedra have been crucial in the success of branch-and-bound algorithms for mixed integer linear programming. Another example, is how the ellipsoid method can be used to prove polynomiality results in combinatorial optimization. For the future, the importance of combinatorial geometry in optimization appears even greater.
In the past 5 years two beautiful geometric algorithms on polyhedra
have been used to prove unexpected new results on the computation of integer programs (both linearly *and* non-linearly constrained). The first is Barvinok's algorithm for polytopes, the second is Graver's bases method on polyhedral cones. I will describe these two algorithms and explain why we can now prove theorems that were beyond our reach before. I will also describe attempts to turn these two algorithms into practical computation, not just in theoretical results.
This a nice story collecting results contained in several papers joint work with various subsets of the following people: R. Hemmecke, M. Koeppe, S. Onn, U. Rothblum, and R. Weismantel.