Polymath3 (PHC6): The Polynomial Hirsch Conjecture – A Topological Approach

This is a new polymath3 research thread. Our aim is to tackle the polynomial Hirsch conjecture which asserts that there is a polynomial upper bound for the diameter of graphs of d-dimensional polytopes with n facets. Our research so far was devoted to an abstract combinatorial setting. We studied an appealing conjecture by Nicolai Hahnle and considered an even more general abstraction proposed by Yury Volvovskiy. Comments towards this abstract conjecture are most welcome!

Here, I would like to mention a topological approach which follows a result that was discovered independently by Tamon Stephen and Hugh Thomas in their paper An Euler characteristic proof that 4-prismatoids have width at most 4,
and by Paco Santos in his paper Embedding a pair of graphs in a surface, and the width of 4-dimensional prismatoids. This post is based on a discussion with Paco Santos at Oberwolfach.

Two maps on a two dimensional Sphere

Theorem: Given a red map and a blue map drawn in general position on S^2 there is an intersection point of two edges of different colors which is adjacent (in the refined map) to a red vertex and to a blue vertex.

Blue and black maps

There are two proofs for the theorem. The proof by Stephen and Thomas uses an Euler characteristic argument. The proof by Santos applies a connectivity argument. Both papers are short and elegant. Both papers point out that the result does not hold for maps on a torus.

Santos’ counterexample to the Hirsch conjecture is based on showing that a direct extension of this result to maps in three dimensions fails. (Even for two maps coming from fans based on polytopes.) Of course, first Paco found his counterexample and then the two-map theorem was found in response to his question  of whether one can find in dimension four counterexamples of the kind he presented in dimension five.

The theorem by Santos, Stephen, and Thomas is very elegant. The proofs are simple but far from obvious and it seems to me that the result will find interesting applications. Its elegance and depth reminded me of Anton Klyachko’s car crash theorem.

A topological question in high dimensions

Now we are ready to present a higher-dimensional analog:

Tentative Conjecture: Let M_1 be a red map and let  M_2 be a blue map drawn in general position on S^{n}, and let $M$ be their common refinement.  There is a vertex w of M a blue vertex v \in M_1, a red vertex u \in M_2 and two faces F,~G \in M such that 1) v,w \in F, 2) w,u \in G, and 3) \dim F + \dim G =n.

A simple (but perhaps not the most general) setting in which to ask this question is with regard to the red and blue maps  coming from red and blue polyhedral fans associated to red and blue convex polytopes. The common refinement will be the fan obtained by taking all intersections of cones, one from the first fan and one from the second.

(Perhaps when n=2k we can even guarantee that \dim F=\dim G=k.)

Why the tentative conjecture implies that the diameter is polynomial

An affirmative answer to this conjecture will lead to a bound of the form dn for the graph of d-polytopes with n facets.

Here is why:

– It is known that the diameter of every polytope with n facets and dimension d is bounded above by the “length” of a Dantzig figure with 2n-2d facets and n-d vertices.

Here a Dantzig figure is a simple polytope of dimension D with 2D facets and two complementary vertices. (i.e., two vertices such that each vertex lies in half of the facets, and they do not belong to any common facet).

The length of the Dantzig figure is the graph distance between these two vertices. This is the classical “d-step theorem” of Klee and Walkup, 1967.

– The length of a Dantzig figure of dimension d is the same as the minimum distance between blue and red vertices in a pair of two maps in the (d-2)-sphere, with d cells each.

– Our tentative conjecture implies, by dimension on d, that the minimum distance between blue and red vertices in a pair of maps in the d-sphere and with n cells is bounded above by (essentially) nd. (n cells means “cells of the blue map plus cells of the red map”, not “cells of the common refinement”).

The abstract setting and other approaches

More comments, ideas, and updates on the abstract setting are of course very welcome Also very welcome are other approaches to the polynomial Hirsch conjecture, and discussion of related problems.

An example showing that the theorem fail for blue and red maps on a torus.