Bio
Fritz visted during October 22-28, 2012.
ARC Colloquium
October 22, 2012
Klaus 1116W
Title: Diameter of Polyhedra: Abstractions, new upper bounds and open problems
Abstract: One of the most prominent mysteries in convex geometry is the question whether the diameter of a polyhedron is bounded by a polynomial in the number of facets. The gap between the best known lower bound (linear) and the best known upper bound (n^{log d} by Kalai and Kleitman) is impressive.
After Francisco Santos refuted the classical Hirsch conjecture in 2010, the polynomial Hirsch conjecture, stating that the answer to the question above is "Yes", has received considerable attention. In this talk I present the best known bounds mentioned above in a very simple abstract setting that does not involve any geometry. The polynomial Hirsch conjecture is also open in this abstract setting. I furthermore show polynomial upper bounds on the diameter of polyhedra that are defined by matrices with small sub-determinants and close with open problems.
Bio: Friedrich Eisenbrand's main research interests lie in the field of discrete optimization, in particular in algorithms and complexity, integer programming, geometry of numbers, and applied optimization. He is best known for his work on efficient algorithms for integer programming in fixed dimension and the theory of cutting planes, which are an important tool to solve large scale industrial optimization problems.
