Connected Layer Families and Diameter of Polyhedra Semester Project

Given a polyhedron, its diameter is defined as the maximum distance between two of its vertices, where the distance between two vertices u and v is the number of edges contained in the shortest u− v path in the skeleton of the polyhedron. Bounding the diameter of a polyhedron with respect to its dimension d and the number n of its facets is one of the most important open problems in convex geometry, a problem about which little is known: the best upper bound is exponential (n ), while we don’t know of any polyhedron with superlinear diameter. In 1957 Hirsch conjectured an upperbound of n − d. Although this bound, called Hirsch bound, holds for many classes of polyhedra, it does not hold in general: after more than fifty years a counter example has been given by Santos ([3]), a polyhedron with dimension 43, 86 facets and diameter more than 43. A polynomial upper bound on the diameter of any polyhedron would imply that a polynomial pivot rule for the simplex method can exist, or it could even suggest such a rule, whereas a class of polyhedra with superpolynomial diameter would rule out this possibility. A promising direction of research is to abstract from the geometric setting and consider simpler combinatorial objects that contain polyhedra as special cases. For the abstract objects with defined notion of diameter, the bound on the abstract diameter would give the bound on the diameter of any polyhedra. One of the most successful applications of this idea is in [1], where the concept of Connected Layer Family is introduced.