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Enumeration of 2-level polytopes. (English) Zbl 1414.05023
A polytope is the convex hull of finitely many points in $$\mathbb{R}^d$$. A hyperplane $$H \subset\mathbb{R}^d$$ is a supporting hyperplane for the polytope $$P$$ if $$P$$ lies entirely on one of the two half spaces defined by $$H$$. A facet of $$P$$ is a set $$P \cap H$$, for some supporting hyperplane $$H$$, that has dimension 1 smaller than that of $$P$$, and a vertex of $$P$$ is a point of the form $$P \cap H$$ for some supporting hyperplane $$H$$.
A polytope $$P$$ is 2-level if, for any facet supporting hyperplane $$H$$, the vertices of $$P$$ that are not on $$H$$ are all in one specific translate of $$H$$. (2-level polytopes are also called compressed.) 2-level polytopes have several applications, e.g., in combinatorial optimization and communication complexity.
The paper under review gives an algorithm to enumerate all 2-level polytopes (up to combinatorial type) in a given dimension, and the authors used their algorithm to build a database of 2-level polytopes of dimension $$\le 7$$. The algorithm is recursive, using the fact that every facet of a 2-level polytope is again 2-level.

##### MSC:
 05A15 Exact enumeration problems, generating functions 52B12 Special polytopes (linear programming, centrally symmetric, etc.) 05C17 Perfect graphs 52B55 Computational aspects related to convexity 68W05 Nonnumerical algorithms 90C22 Semidefinite programming
##### Software:
01poly; 2L_enum; birkhoff faces; Boost; nauty; Traces; uBLAS
Full Text:
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