UNIVERSITÉ LIBRE DE BRUXELLES
Faculté des Sciences Département de Mathématique
Année Académique 2017/2018
Thèse présentée en vue de l’obtention du grade de Docteur en Sciences
TWO LEVEL POLYTOPES:
GEOMETRY AND OPTIMIZATION
Marco Macchia
Jury de thèse:
Prof. Dr. Jean Cardinal (président), Université libre de Bruxelles Prof. Dr. Michele D’Adderio (secrétaire), Université libre de Bruxelles Prof. Dr. Samuel Fiorini (promoteur), Université libre de Bruxelles Prof. Dr. Arnau Padrol, Sorbonne Université
Contents
Summary v Résumé vii Acknowledgements ix Introduction 1 1 Basic notions 9 1.1 Overview 9 1.2 Generalities 10 1.2.1 Polytopes 10 1.2.2 Operations on polytopes 11 1.2.3 Slack matrices 12 1.3 Operations preserving 2-levelness 14 1.4 Families of 2-level polytopes 15 1.4.1 Hanner and sequentially Hanner polytopes 15 1.4.2 Stable set polytope of perfect graphs 16 1.4.3 2-level polytopes via totally unimodular matrices 18 1.4.4 2-level polytopes via balanced matrices 18 1.4.5 2-level polytopes via perfect matrices 18 1.4.6 Further families of 2-level polytopes 19 1.5 Embeddings of 2-level polytopes 20 1.5.1 Simplicial cores 20 1.5.2 Slack embedding 22 1.5.3 H- and V-embeddings 23 1.6 Characterizations of 2-level polytopes 26 2 Enumeration of 2-level polytopes 332.1 Overview 33
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2.2 The enumeration algorithm 35 2.2.1 Combining embeddings 36 2.2.2 Discrete convexity 37 2.2.3 Incompatibilities from the base 39 2.2.4 Moore families 40 2.2.5 Pseudocode 41 2.3 Reductions of ground set and complete family 41 2.3.1 Removing points causing incompatibilities 43 2.3.2 Translating the candidate set 44 2.3.3 Complete family for the intersection 48 2.4 Closure operators 49 2.4.1 Discrete convex hull operator 51 2.4.2 Incompatibility operator 51 2.4.3 Closure operator for the intersection 52 2.5 Exploiting symmetries of the base 53 2.5.1 Invariance of Xcompand closed sets 55 2.5.2 Restricting to the �nal ground set 57 2.6 Implementation 60 2.6.1 Storing and comparing 2-level polytopes 60 2.6.2 Testing for 2-levelness 62 2.6.3 Generating the complete family 64 2.6.4 Further optimizations 64 2.7 Experimental results 65 2.7.1 Computational times 66 2.7.2 Parallelization of the computation 67 2.7.3 More statistics 67 3 Bounds for 2-level polytopes, cones, con�gurations 69
3.1 Overview 69
C������� iii
4 Prescribing facets 83
4.1 Overview 83
4.2 Structure of sequentially Hanner polytopes 85 4.3 Strong separation property and suspensions 89 4.4 Prescribing a sequentially Hanner facet 91
4.4.1 Embedding 93
4.4.2 Polyhedral subdivision 94 4.4.3 Bidirected graph 96 4.4.4 Cubical �bers 98 4.4.5 Compatibility �bers/quasi-suspensions 99 5 Extension complexity of STAB of bipartite graphs 113
5.1 Overview 113
5.2 Rectangle covers and fooling sets 116 5.3 An improved upper bound 118 5.4 An improved lower bound 119 5.5 A small rectangle cover of the special entries 124
6 Open questions 129
6.1 Number of 2-level polytopes 129 6.2 Number of faces of a 2-level polytope 130 6.3 Complexity of recognizing 2-level polytopes 131 6.4 Extension complexity of 2-level polytopes 132
