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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
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