Mathematical Programming Computation

, Volume 11, Issue 1, pp 173–210 | Cite as

Enumeration of 2-level polytopes

  • Adam Bohn
  • Yuri Faenza
  • Samuel Fiorini
  • Vissarion Fisikopoulos
  • Marco MacchiaEmail author
  • Kanstantsin Pashkovich
Full Length Paper


A (convex) polytope P is said to be 2-level if for each hyperplane H that supports a facet of P, the vertices of P can be covered with H and exactly one other translate of H. The study of these polytopes is motivated by questions in combinatorial optimization and communication complexity, among others. In this paper, we present the first algorithm for enumerating all combinatorial types of 2-level polytopes of a given dimension d, and provide complete experimental results for \(d \leqslant 7\). Our approach is inductive: for each fixed \((d-1)\)-dimensional 2-level polytope \(P_0\), we enumerate all d-dimensional 2-level polytopes P that have \(P_0\) as a facet. This relies on the enumeration of the closed sets of a closure operator over a finite ground set. By varying the prescribed facet \(P_0\), we obtain all 2-level polytopes in dimension d.


Polyhedral computation Polyhedral combinatorics Optimization Formal concept analysis Algorithm engineering 

Mathematics Subject Classification

05A15 05C17 52B12 52B55 68W05 90C22 



We acknowledge support from the following research grants: ERC grant FOREFRONT (grant agreement no. 615640) funded by the European Research Council under the EU’s 7th Framework Programme (FP7/2007-2013), Ambizione grant PZ00P2 154779 Tight formulations of 0-1 problems funded by the Swiss National Science Foundation, the research grant Semidefinite extended formulations (Semaphore 14620017) funded by F.R.S.-FNRS, and the ARC grant AUWB-2012-12/17-ULB2 COPHYMA funded by the French community of Belgium.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and The Mathematical Programming Society 2018

Authors and Affiliations

  • Adam Bohn
    • 1
  • Yuri Faenza
    • 2
  • Samuel Fiorini
    • 1
  • Vissarion Fisikopoulos
    • 1
  • Marco Macchia
    • 1
    Email author
  • Kanstantsin Pashkovich
    • 3
  1. 1.Université libre de BruxellesBrusselsBelgium
  2. 2.IEOR DepartmentColumbia UniversityNew YorkUSA
  3. 3.C&O DepartmentUniversity of WaterlooWaterlooCanada

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