A quasi-polynomial bound for the diameter of graphs of polyhedra

The diameter of the graph of a d-dimensional polyhedron with n facets is at most nlog d+2 Let P be a convex polyhedron. The graph of P denoted by G(P ) is an abstract graph whose vertices are the extreme points of P and two vertices u and v are adjacent if the interval [v, u] is an extreme edge (= 1-dimensional face) of P . The diameter of the graph of P is denoted by δ(P ). Let ∆(d, n) be the maximal diameter of the graphs of d-dimensional polyhedra P with n facets. (A facet is a (d− 1)-dimensional face.) Thus, P is the set of solutions of n linear inequalities in d variables. It is an old standing problem to determine the behavior of the function ∆(d, n). The value of ∆(d, n) is a lower bound for the number of iterations needed for Dantzig’s simplex algorithm for linear programming with any pivot rule. In 1957 Hirsch conjectured [2] that ∆(d, n) ≤ n−d. Klee and Walkup [6] showed that the Hirsch conjecture is false for unbounded polyhedra. They proved that for n ≥ 2d,∆(d, n) ≥ n − d + [d/5]. This is the best known lower bound for ∆(d, n). The statement of the Hirsch conjecture for bounded polyhedra is still open. For a recent survey on the Hirsch conjecture and its relatives, see [5]. In 1967 Barnette proved [1, 3] that ∆(d, n) ≤ n3. An improved upper bound, ∆(d, n) ≤ n2, was proved in 1970 by Larman [7]. Barnette’s and Larman’s bounds are linear in n but exponential in the dimension d. In 1990 the first author [4] proved a subexponential bound ∆(d, n) ≤ 2 √ (n−d) . The purpose of this paper is to announce and to give a complete proof of a quasipolynomial upper bound for ∆(d, n). Such a bound was proved by the first author in March 1991. The proof presented here is a substantial simplification that was subsequently found by the second author. See [4] for the original proof and related results. The existence of a polynomial (or even linear) upper bound for ∆(d, n) is still open. Recently, the first author found a randomized pivot rule for linear programming which requires an expected n √ d (or less) arithmetic operations for every linear programming problem with d variables and n constraints. 1991 Mathematics Subject Classification. Primary 52A25, 90C05. Received by the editors July 1, 1991 The first author was supported in part by a BSF grant by a GIF grant. The second author was supported by an AFOSR grant c ©1992 American Mathematical Society 0273-0979/92 $1.00 + $.25 per page