Lectures on 0/1-Polytopes

These lectures on the combinatorics and geometry of 0/1-polytopes are meant as an introduction and invitation. Rather than heading for an extensive survey on 0/1polytopes I present some interesting aspects of these objects; all of them are related to some quite recent work and progress. 0/1-polytopes have a very simple definition and explicit descriptions; we can enumerate and analyze small examples explicitly in the computer (e. g. using polymake). However, any intuition that is derived from the analysis of examples in “low dimensions” will miss the true complexity of 0/1-polytopes. Thus, in the following we will study several aspects of the complexity of higher-dimensional 0/1-polytopes: the doubly-exponential number of combinatorial types, the number of facets which can be huge, and the coefficients of defining inequalities which sometimes turn out to be extremely large. Some of the effects and results will be backed by proofs in the course of these lectures; we will also be able to verify some of them on explicit examples, which are accessible as a polymake database.

[1]  R. Richardson.  The International Congress of Mathematicians , 1932, Science.

[2]  J. Williamson,et al.  Determinants whose Elements are 0 and 1 , 1946 .

[3]  Michel Balinski,et al.  On the graph structure of convex polyhedra in n-space , 1961 .

[4]  Determinants with Elements ± 1 , 1967 .

[5]  On a class of (0,1) matrices with vanishing determinants , 1967 .

[6]  Saburo Muroga,et al.  Threshold logic and its applications , 1971 .

[7]  Vasek Chvátal,et al.  Edmonds polytopes and a hierarchy of combinatorial problems , 1973, Discret. Math..

[8]  A. Mukhopadhyay,et al.  On the probability that the determinant of an n x n matrix over a finite field vanishes , 1984, Discret. Math..

[9]  Eugene L. Lawler,et al.  Traveling Salesman Problem , 2016 .

[10]  György Elekes,et al.  A geometric inequality and the complexity of computing volume , 1986, Discret. Comput. Geom..

[11]  Ali Ridha Mahjoub,et al.  On the cut polytope , 1986, Math. Program..

[12]  Z. Fiiredi.  Random Polytopes in the d-Dimensional Cube , 1986 .

[13]  Zoltán Füredi.  Random polytopes in thed-dimensional cube , 1986, Discret. Comput. Geom..

[14]  Andrew M. Odlyzko,et al.  On subspaces spanned by random selections of plus/minus 1 vectors , 1988, Journal of combinatorial theory. Series A.

[15]  Denis Naddef,et al.  The hirsch conjecture is true for (0, 1)-polytopes , 1989, Math. Program..

[16]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[17]  J. H. E. Cohn.  On Determinants with Elements ±1, II , 1989 .

[18]  H. P. Williams.  THEORY OF LINEAR AND INTEGER PROGRAMMING (Wiley-Interscience Series in Discrete Mathematics and Optimization) , 1989 .

[19]  Warren D. Smith.  Studies in computational geometry motivated by mesh generation , 1989 .

[20]  N. Biggs.  GEOMETRIC ALGORITHMS AND COMBINATORIAL OPTIMIZATION: (Algorithms and Combinatorics 2) , 1990 .

[21]  P. Erdos,et al.  Collected Papers of Paul Turan , 1990 .

[22]  R. Blind,et al.  Convex polytopes without triangular faces , 1990 .

[23]  Caterina De Simone,et al.  The cut polytope and the Boolean quadric polytope , 1990, Discret. Math..

[24]  Mark Haiman,et al.  A simple and relatively efficient triangulation of then-cube , 1991, Discret. Comput. Geom..

[25]  Tomás Feder,et al.  Balanced matroids , 1992, STOC '92.

[26]  Milena Mihail.  On the Expansion of Combinatorial Polytopes , 1992, MFCS.

[27]  Michel Deza,et al.  The cut cone III: On the role of triangle facets , 1992, Graphs Comb..

[28]  Hyperebenen in Hyperkuben - Eine Klassifizierung und Quantifizierung , 1992 .

[29]  Michael R. Anderson,et al.  A triangulation of the 6-cube with 308 simplices , 1993, Discret. Math..

[30]  Robert B. Hughes.  Minimum-cardinality triangulations of the d-cube for d=5 and d=6 , 1993, Discret. Math..

[31]  Robert B. Hughes,et al.  Lower bounds on cube simplexity , 1994, Discret. Math..

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

[33]  Johan Håstad,et al.  On the Size of Weights for Threshold Gates , 1994, SIAM J. Discret. Math..

[34]  Rekha R. Thomas,et al.  Gröbner bases and triangulations of the second hypersimplex , 1995, Comb..

[35]  E. Szemerédi,et al.  On the probability that a random ±1-matrix is singular , 1995 .

[36]  Franz Aurenhammer,et al.  Classifying Hyperplanes in Hypercubes , 1996, SIAM J. Discret. Math..

[37]  Michael R. Anderson,et al.  Simplexity of the cube , 1996, Discret. Math..

[38]  Victor Klee,et al.  Largest j-simplices in d-cubes: Some relatives of the hadamard maximum determinant problem , 1996 .

[39]  Louis J. Billera,et al.  All 0–1 polytopes are traveling salesman polytopes , 1996, Comb..

[40]  Michel Deza,et al.  Geometry of cuts and metrics , 2009, Algorithms and combinatorics.

[41]  Günter M. Ziegler,et al.  Extremal Properties of 0/1-Polytopes , 1997, Discret. Comput. Geom..

[42]  Noga Alon,et al.  Anti-Hadamard Matrices, Coin Weighing, Threshold Gates, and Indecomposable Hypergraphs , 1997, J. Comb. Theory, Ser. A.

[43]  Michael G. Neubauer,et al.  The maximum determinant of ± 1 matrices , 1997 .

[44]  V. Klee,et al.  A proof of the strict monotone 4-step conjecture , 1998 .

[45]  R. Bixby,et al.  On the Solution of Traveling Salesman Problems , 1998 .

[46]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[47]  Friedrich Eisenbrand,et al.  Bounds on the Chvátal Rank of Polytopes in the 0/1-Cube , 1999, IPCO.

[48]  David P. Robbins,et al.  On the Volume of the Polytope of Doubly Stochastic Matrices , 1999, Exp. Math..

[49]  Friedrich Eisenbrand,et al.  On the Chvátal Rank of Polytopes in the 0/1 Cube , 1999, Discret. Appl. Math..

[50]  Michael Joswig,et al.  polymake: a Framework for Analyzing Convex Polytopes , 2000 .

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

[52]  Volker Kaibel,et al.  Simple 0/1-Polytopes , 2000, Eur. J. Comb..

[53]  Jesús A. De Loera,et al.  Minimal Simplicial Dissections and Triangulations of Convex 3-Polytopes , 2000, Discret. Comput. Geom..

[54]  Günter Rote,et al.  Upper Bounds on the Maximal Number of Facets of 0/1-Polytopes , 2000, Eur. J. Comb..

[55]  Warren D. Smith.  A Lower Bound for the Simplexity of then-Cube via Hyperbolic Volumes , 2000, Eur. J. Comb..