## Abstract

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.

Original language | English |
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Pages (from-to) | 69-71 |

Number of pages | 3 |

Journal | Information Processing Letters |

Volume | 100 |

Issue number | 2 |

DOIs | |

State | Published - 31 Oct 2006 |

Externally published | Yes |

## Keywords

- Computational geometry
- Linear programming
- Polyhedron
- Upper bounds