UPPER-BOUNDS FOR THE DIAMETER AND HEIGHT OF GRAPHS OF CONVEX POLYHEDRA

Let DELTA(d, n) be the maximum diameter of the graph of a d-dimensional polyhedron P with n-facets. It was conjectured by Hirsch in 1957 that DELTA(d, n) depends linearly on n and d. However, all known upper bounds for DELTA(d, n) were exponential in d. We prove a quasi-polynomial bound DELTA(d, n) less-than-or-equal-to n2logd+3. Let P be a d-dimensional polyhedron with n facets, let phi be a linear objective function which is bounded on P and let upsilon be a vertex of P. We prove that in the graph of P there exists a monotone path leading from upsilon to a vertex with maximal phi-value whose length is at most n2 square-root n.