On the Vertices, Facets, and Graph Diameter of Delta-Modular Polyhedra

This short tutorial is devoted to delta-modular polyhedra and their properties. A polyhedron defined by a system $(A x \leq b)$ with integer $(A, b)$ is called delta-modular if the rank-order sub-determinants of the matrix $(A)$ are bounded in absolute value by delta. We will present results on the number of vertices and facets of such polyhedra, with special attention given to the problem of the diameter of their graph. In the latter case, we will explain a probabilistic technique for obtaining the best-known bounds on the diameter and dedicate time to open questions that may lead to improved diameter bounds.