A Diameter Problem (2)

2. The connection with Hirsch’s Conjecture

The Hirsch Conjecture asserts that the diameter of the graph G(P) of a d-polytope P with n facets is at most n-d. Not even a polynomial upper bound for the diameter in terms of d and n is known. Finding good upper bounds for the diameter of graphs of d-polytopes is one of the central open problems in the study of convex polytopes. If d is fixed then a linear bound in n is known, and the best bound in terms of d and n is . We will come back to these results later.

One basic fact to remember is that for every d-polytope P, G(P) is a connected graph. As a matter of fact, a theorem of Balinski asserts that G(P)$ is d-connected.

The combinatorial diameter problem I mentioned in an earlier post (and which is repeated below) is closely related. Let me now explain the connection. 

Let P be a simple d-polytope. Suppose that P is determined by n inequalities, and that each inequality describes a facet of P. Now we can define a family of subsets of {1,2,…,n} as follows. Let be the n inequalities defining the polytope P, and let be the n corresponding facets. Every vertex v of P belongs to precisely d facets (this is equivalent to P being a simple polytope). Let be the indices of the facets containing v, or, equivalently, the indices of the inequalities which are satisfied as equalities at v. Now, let be the family of all sets for all vertices of the polytope P.

The following observations are easy.  

(1) Two vertices v and w of P are adjacent in the graph of P if and only if . Therefore, .

(2)  If A is a set of indices. The vertices v of P such that are precisely the set of vertices of a lower dimensional face of P. This face is described by all the vertices of P which satisfies all the inequalities indexed by , or equivalently all vertices in P which belong to the intersection of the facets for .

Therefore, for every if is not empty the graph is connected – this graph is just the graph of some lower dimensional polytope. This was the main assumption in our abstract problem.

Remark: It is known that the assertion of the Hirsch Conjecture fails for the abstract setting. There are examples of families where the diameter is as large as n-(4/5)d.

1. A Diameter problem for families of sets

Let me draw your attention to the following problem:

Consider a family of subsets of size d of the set N={1,2,…,n}.

Associate to a graph as follows: The vertices of  are simply the sets in . Two vertices and are adjacent if .

For a subset let denote the subfamily of all subsets of which contain . 

MAIN ASSUMPTION: Suppose that for every for which is not empty is connected.

MAIN QUESTION:   How large can the diameter of be in terms of and .