A geometric lower bound on the extension complexity of polytopes based on the f-vector

Abstract A linear extension of a polytope Q is any polytope which can be mapped onto Q via an affine transformation. The extension complexity of a polytope is the minimum number of facets of any linear extension of this polytope. In general, computing the extension complexity of a given polytope is difficult. The extension complexity is also equal to the nonnegative rank of any slack matrix of the polytope. In this paper, we introduce a new geometric lower bound on the extension complexity of a polytope, i.e., which relies only on the knowledge of some of its geometric characteristics. It is based on the monotone behaviour of the f -vector of polytopes under affine maps. We present numerical results showing that this bound can improve upon existing geometric lower bounds, and provide a generalization of this lower bound for the nonnegative rank of any matrix.

[1]  G. C. Shephard,et al.  Convex Polytopes , 1969, The Mathematical Gazette.

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

[3]  Louis J. Billera,et al.  Counting Faces and Chains in Polytopes and Posets , 1983 .

[4]  Gil Kalai,et al.  On low-dimensional faces that high-dimensional polytopes must have , 1990, Comb..

[5]  Alfred V. Aho,et al.  On notions of information transfer in VLSI circuits , 1983, STOC.

[6]  Sanjeev Arora,et al.  Computing a nonnegative matrix factorization -- provably , 2011, STOC '12.

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

[8]  Stefan Weltge,et al.  Computing The Extension Complexities of All 4-Dimensional 0/1-Polytopes , 2014, 1406.4895.

[9]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[10]  Joel E. Cohen,et al.  Nonnegative ranks, decompositions, and factorizations of nonnegative matrices , 1993 .

[11]  Thomas Rothvoß,et al.  The matching polytope has exponential extension complexity , 2013, STOC.

[12]  Nicolas Gillis.  Nonnegative matrix factorization : complexity, algorithms and applications , 2011 .

[13]  P. McMullen.  The maximum numbers of faces of a convex polytope , 1970 .

[14]  Nicolas Gillis,et al.  Heuristics for exact nonnegative matrix factorization , 2014, J. Glob. Optim..

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

[16]  Arne Bronsted,et al.  A dual proof of the upper bound conjecture for convex polytopes. , 1980 .

[17]  Raimund Seidel,et al.  The Upper Bound Theorem for Polytopes: an Easy Proof of Its Asymptotic Version , 1995, Comput. Geom..

[18]  Stephen A. Vavasis,et al.  On the Complexity of Nonnegative Matrix Factorization , 2007, SIAM J. Optim..

[19]  Louis J. Billera,et al.  Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets , 1985 .

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

[21]  Samuel Fiorini,et al.  Combinatorial bounds on nonnegative rank and extended formulations , 2011, Discret. Math..

[22]  Michel X. Goemans,et al.  Smallest compact formulation for the permutahedron , 2015, Math. Program..

[23]  Margaret M. Bayer,et al.  Combinatorial Aspects of Convex Polytopes , 1993 .

[24]  James Worrell,et al.  On Restricted Nonnegative Matrix Factorization , 2016, ICALP.

[25]  Michael T. Goodrich,et al.  On the Complexity of Optimization Problems for 3-dimensional Convex Polyhedra and Decision Trees , 1997, Comput. Geom..

[26]  Pablo A. Parrilo,et al.  Self-scaled bounds for atomic cone ranks: applications to nonnegative rank and cp-rank , 2014, Math. Program..