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

##### MPG-Autoren
http://pubman.mpdl.mpg.de/cone/persons/resource/persons44374

Elbassioni,  Khaled
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons44944

Lotker,  Zvi
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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##### Zitation

Elbassioni, K., Lotker, Z., & Seidel, R. (2006). Upper bound on the number of vertices of polyhedra with 0, 1-constraint matrices. Information Processing Letters, 100, 69-71.

In this note we give upper bounds for the number of vertices of the polyhedron $P(A,b) = \{x \in Rd: Ax &lt; b\}$ when the $m \times d$ 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 $Cd$ vertices. These bounds are consequences of a more general theorem that the number of vertices of $P(A,b)$ is at most $d! ċ W/D$, 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 $d \times d$ subdeterminant of $A$.