Abstract
We propose the first algorithm for enumerating all combinatorial types of 2-level polytopes of a given dimension d, and provide complete experimental results for \(d \leqslant 6\). Our approach is based on the notion of a simplicial core, that allows us to reduce the problem to the enumeration of the closed sets of a discrete closure operator, along with some convex hull computations and isomorphism tests.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Aichholzer, O.: Extremal properties of 0/1-polytopes of dimension 5. In: Ziegler, G., Kalai, G. (eds.) Polytopes - Combinatorics and Computation, pp. 111–130. Birkhäuser (2000)
Akutsu, T.: On determining the congruence of point sets in d dimensions. Computational Geometry 9(4), 247–256 (1998)
Beyer, S.: Comprehensive perl archive network: Bit-vector-7.4 (2014). http://search.cpan.org/~stbey/Bit-Vector-7.4
Chvátal, V.: On certain polytopes associated with graphs. J. Combinatorial Theory Ser. B 18, 138–154 (1975)
De Loera, J., Rambau, J., Santos, F.: Triangulations. Algorithms and Computation in Mathematics, vol. 25 (2010)
Ganter, B., Reuter, K.: Finding all closed sets: A general approach. Order 8(3), 283–290 (1991) (English)
Gawrilow, E., Joswig, M.: Polymake: an approach to modular software design in computational geometry. In: International Symposium on Computational Geometry (SOCG), pp. 222–231. ACM Press (2001)
Gouveia, J., Parrilo, P., Thomas, R.: Theta bodies for polynomial ideals. SIAM Journal on Optimization 20(4), 2097–2118 (2010)
Gouveia, J., Robinson, R., Thomas, R.: Polytopes of minimum positive semidefinite rank. Discrete & Computational Geometry 50(3), 679–699 (2013)
Grande, F., Sanyal, R.: Theta rank, levelness, and matroid minors, arXiv preprint, arXiv:1408.1262 (2014)
Haase, C.: Lattice polytopes and unimodular triangulations, Ph.D. thesis, Technical University of Berlin (2000)
Hanner, O.: Intersections of translates of convex bodies. Mathematica Scandinavica 4, 65–87 (1956)
Hansen, A.: On a certain class of polytopes associated with independence systems. Mathematica Scandinavica 41, 225–241 (1977)
Kalai, G.: The number of faces of centrally-symmetric polytopes. Graphs and Combinatorics 5(1), 389–391 (1989)
Kuznetsov, S., Obiedkov, S.: Comparing performance of algorithms for generating concept lattices. Journal of Experimental and Theoretical Artificial Intelligence 14, 189–216 (2002)
Paffenholz, A.: Faces of Birkhoff polytopes. Electronic Journal of Combinatorics 22 (2015)
Sanyal, R., Werner, A., Ziegler, G.: On Kalai’s conjectures concerning centrally symmetric polytopes. Discrete & Computational Geometry 41(2), 183–198 (2009)
Stanley, R.: Decompositions of rational convex polytopes. Annals of Discrete Mathematics, 333–342 (1980)
Stanley, R.: Two poset polytopes. Discrete & Computational Geometry 1(1), 9–23 (1986)
Sullivant, S.: Compressed polytopes and statistical disclosure limitation. Tohoku Mathematical Journal, Second Series 58(3), 433–445 (2006)
Ziegler, G.: Lectures on polytopes, vol. 152, Springer Science & Business Media (1995)
Ziegler, G.: Lectures on 0/1-polytopes. In: Kalai, G., Ziegler, G. (eds.) Polytopes - Combinatorics and Computation. DMV Seminar, vol. 29, pp. 1–41. Springer, Birkhäuser, Basel (2000)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bohn, A., Faenza, Y., Fiorini, S., Fisikopoulos, V., Macchia, M., Pashkovich, K. (2015). Enumeration of 2-Level Polytopes. In: Bansal, N., Finocchi, I. (eds) Algorithms - ESA 2015. Lecture Notes in Computer Science(), vol 9294. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48350-3_17
Download citation
DOI: https://doi.org/10.1007/978-3-662-48350-3_17
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-48349-7
Online ISBN: 978-3-662-48350-3
eBook Packages: Computer ScienceComputer Science (R0)