Enumeration of 2-level polytopes

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 ≤ 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.

[1]  G. Cornuéjols,et al.  Combinatorial optimization : packing and covering , 2001 .

[2]  Richard P. Stanley,et al.  Two poset polytopes , 1986, Discret. Comput. Geom..

[3]  Bernhard Ganter,et al.  Formal Concept Analysis: Mathematical Foundations , 1998 .

[4]  G. Ziegler.  Lectures on Polytopes , 1994 .

[5]  Grigoriy Blekherman,et al.  Nonnegative Polynomials and Sums of Squares , 2010, 1010.3465.

[6]  Rekha R. Thomas,et al.  Four-dimensional polytopes of minimum positive semidefinite rank , 2017, J. Comb. Theory, Ser. A.

[7]  Sonoko Moriyama,et al.  Complete Enumeration of Small Realizable Oriented Matroids , 2012, CCCG.

[8]  B. Reed,et al.  Recent advances in algorithms and combinatorics , 2003 .

[9]  Michael Joswig,et al.  Polymake: an approach to modular software design in computational geometry , 2001, SCG '01.

[10]  Sergei O. Kuznetsov,et al.  Comparing performance of algorithms for generating concept lattices , 2002, J. Exp. Theor. Artif. Intell..

[11]  Alfonso Cevallos,et al.  On vertices and facets of combinatorial 2-level polytopes , 2016, ISCO.

[12]  O. Hanner,et al.  Intersections of translates of convex bodies , 1956 .

[13]  A. B. Hansen.  On a certain class of polytopes associated with independence systems. , 1977 .

[14]  Brendan D. McKay,et al.  Practical graph isomorphism, II , 2013, J. Symb. Comput..

[15]  Francesco Grande,et al.  Theta rank, levelness, and matroid minors , 2014, J. Comb. Theory, Ser. B.

[16]  B. Ganter,et al.  Finding all closed sets: A general approach , 1991 .

[17]  Andreas Paffenholz.  Faces of Birkhoff Polytopes , 2015, Electron. J. Comb..

[18]  G. Ziegler.  Lectures on 0/1-Polytopes , 1999, math/9909177.

[19]  Oswin Aichholzer,et al.  Extremal Properties of 0/1-Polytopes of Dimension 5 , 2000 .

[20]  Günter M. Ziegler,et al.  On Kalai’s Conjectures Concerning Centrally Symmetric Polytopes , 2007, Discret. Comput. Geom..

[21]  Seth Sullivant.  Compressed polytopes and statistical disclosure limitation , 2004 .

[22]  Francesco Grande,et al.  Many 2-Level Polytopes from Matroids , 2014, Discret. Comput. Geom..

[23]  Francesco Grande,et al.  On k-level matroids: geometry and combinatorics , 2015 .

[24]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[25]  Michael E. Saks,et al.  Lattices, mobius functions and communications complexity , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[26]  Rekha R. Thomas,et al.  Polytopes of Minimum Positive Semidefinite Rank , 2012, Discret. Comput. Geom..

[27]  Rekha R. Thomas,et al.  Theta Bodies for Polynomial Ideals , 2008, SIAM J. Optim..

[28]  Nicolas Gillis,et al.  On the Geometric Interpretation of the Nonnegative Rank , 2010, 1009.0880.

[29]  M. Bauer,et al.  Triangulations , 1996, Discret. Math..

[30]  M. Yannakakis.  Expressing combinatorial optimization problems by linear programs , 1991, Symposium on the Theory of Computing.

[31]  V. Chvátal.  On certain polytopes associated with graphs , 1975 .

[32]  Gil Kalai,et al.  The number of faces of centrally-symmetric polytopes , 1989, Graphs Comb..

[33]  Kanstantsin Pashkovich,et al.  Extended Formulations for Combinatorial Polytopes , 2012 .

[34]  Tatsuya Akutsu.  On determining the congruence of point sets in d dimensions , 1998, Comput. Geom..

[35]  Rekha R. Thomas,et al.  Which nonnegative matrices are slack matrices , 2013, 1303.5670.

[36]  Richard P. Stanley,et al.  Decompositions of Rational Convex Polytopes , 1980 .