Upper bound on the number of vertices of polyhedra with 0, 1-constraint matrices

In this note we give upper bounds for the number of vertices of the polyhedron P (A, b) = {x ∈ R^d : A x ≤ b} when the m × d constraint matrix A is subjected to certain restriction. For instance, if A is a 0/1-matrix, then there can be at most d! vertices and this bound is tight, or if the entries of A are non-negative integers so that each row sums to at most C, then there can be at most C^d vertices. These bounds are consequences of a more general theorem that the number of vertices of P (A, b) is at most d ! ṡ W / D, where W is the volume of the convex hull of the zero vector and the row vectors of A, and D is the smallest absolute value of any non-zero d × d subdeterminant of A.