Upper Bound on the Number of Vertices of Polyhedra with $0,1$-Constraint Matrices

TL;DR

This paper establishes that the maximum number of vertices of a polyhedron defined by a 0-1 constraint matrix in d-dimensional space is at most d factorial, providing a tight upper bound.

Contribution

It proves a new upper bound of d! on the number of vertices for polyhedra with 0-1 constraint matrices, advancing understanding of their combinatorial complexity.

Findings

Maximum vertices bounded by d! for polyhedra with 0-1 matrices

Provides a tight upper bound on polyhedral vertices

Enhances combinatorial understanding of 0-1 polyhedra

Abstract

In this note we show that the maximum number of vertices in any polyhedron $P=\{x\in \mathbb{R}^d : Ax\leq b\}$ with $0,1$-constraint matrix $A$ and a real vector $b$ is at most $d!$.