Diss. ETH No. 14335 A Graph Theoretical Approach for Reconstruction and Generation of Oriented Matroids A dissertation submitted to the SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH for the degree of DOCTOR OF MATHEMATICS presented by LUKAS FINSCHI dipl. Math. ETH born 1972 March 18 citizen of Safien GR accepted on the recommendation of Prof. Hans-Jakob L¨uthi, examiner Prof. Komei Fukuda, co-examiner Prof. G¨unter M. Ziegler, co-examiner 2001 Contents Acknowledgements xiii Abstract xv Zusammenfassung xvii Structure Diagram xix 0 Introduction 1 Motivation and Overview 3 0 An Introduction to Oriented Matroids 15 0.1AFirstTourofOrientedMatroids..................... 15 0.2CovectorAxioms.............................. 21 0.3Matroids................................... 23 0.4Minors.................................... 28 0.5 Duality . .............................. 32 0.6Cocircuits.................................. 37 0.7 Topes and the Big Face Lattice ....................... 41 0.8OrientedMatroidProgramming...................... 46 0.9BasisOrientationsandChirotopes..................... 48 iv CONTENTS I Reconstruction 53 1 Topes and Tope Graphs 55 1.1 Introduction and Problem Statements .................... 55 1.2PropertiesofTopesGraphs......................... 58 1.3 Separability of Uncut Topes . .................... 61 1.4 Orientation Reconstruction . .................... 66 1.5 Face Reconstruction from Topes . .................... 68 1.6 Construction of Covectors and Topes from Cocircuits . ......... 71 1.7 Algorithmic Characterization of Tope Sets . ............. 72 2 Cocircuits and Cocircuit Graphs 75 2.1 Introduction and Problem Statements .................... 75 2.2 Orientation Reconstruction from Matroid Label . ............. 78 2.3 Reconstruction of Uniform Matroid Labels from Antipodes . 82 2.4 Antipodes in Uniform Cocircuit Graphs . ............. 86 2.5 Characterization of Cocircuit Graphs .................... 89 2.6OpenProblems............................... 90 II Generation 93 3 Generation of Oriented Matroids and Isomorphism Classes 95 3.1 Introduction . ........................... 95 3.2 Duality and the Generation of Isomorphism Classes . ......... 96 3.3 Incremental Method for the Generation of Isomorphism Classes . 99 3.4 Graph Representations in Generation Methods . .............102 4 Tope Graphs and Single Element Extensions 105 4.1 Tope Graphs and Isomorphism Classes of Oriented Matroids . 105 4.2LocalizationsandTopeGraphExtensions.................107 4.3 Reverse Search Method for the Generation of Localizations . 111 CONTENTS v 4.4 Reduction of Multiple Extension Using Isomorphic Signatures . .....113 4.5 Reduction of Multiple Extension Using Maximal Localizations . .....115 5 Cocircuit Graphs and Single Element Extensions 117 5.1CocircuitGraphsandIsomorphismClassesofOrientedMatroids.....117 5.2LocalizationsandCocircuitGraphExtensions...............120 5.3 Two Methods for the Generation of Localizations . ............125 5.4 Backtracking Method for the Generation of Localizations . .....126 III Applications 131 6 A Catalog of Oriented Matroids 133 6.1 Introduction . ..............................133 6.2OrganizationofCatalog...........................134 6.3PropertiesofCatalog............................136 6.4GenerationofCatalog............................138 6.5OverviewofResults.............................141 6.6 Access to Catalog and Examples . ...................144 7 Complete Listing of Point Configurations 147 7.1 Introduction . ..............................147 7.2PointConfigurationsandAcyclicOrientedMatroids...........148 7.3 Generation of Abstract Order Types . ...................150 7.4PolytopesandMatroidPolytopes......................154 7.5AConjectureRelatedtotheSylvester-GallaiTheorem..........156 8 Complete Listing of Hyperplane Arrangements 161 8.1 Introduction . ..............................161 8.2HyperplaneArrangementsandAffineOrientedMatroids.........162 8.3 Generation of Abstract Dissection Types . ............164 vi CONTENTS Bibliography 169 Glossary of Notation 175 Subject Index 177 List of Figures 1 Sphere arrangement . ....................... 3 2 Sphere arrangements of non-uniform and uniform oriented matroids . 6 3 Adjacent regions in sphere arrangement and tope graph . ..... 6 4 Sphere arrangement and cocircuit graph . ............ 7 5 Uncut regions in sphere arrangement ( f 4)............... 9 = 6 ColinecyclesinM-labeledcocircuitgraph................. 10 7 Localizationsoftopegraphandcocircuitgraph.............. 12 8 Signaturesoncolinecyclesinducedbyalocalization........... 13 9 The 3 order types with 4 non-collinear points in R2 ............ 14 10 The 3 dissection types with 3 non-parallel hyperplanes in R2 ....... 14 0.1 Sphere arrangement . ....................... 18 0.2 Pseudosphere arrangement . ....................... 19 0.3 The big face lattice F ............................ 20 ˆ 0.4Diamondproperty.............................. 45 1.1 Adjacent regions in pseudosphere arrangement and tope graph . ..... 56 1.2Topegraphofanorientedmatroidofrank2................ 61 1.3Examplefornon-connectednessintheaffinecase............. 63 1.4 Acycloid whose tope graph is not separable . ............ 64 1.5 Acycloid which is not an oriented matroid but whose tope graph is separable 65 1.6ThetwocasesintheproofofTheorem1.3.1................ 65 2.1 Pseudosphere arrangement and cocircuit graph . ............ 76 viii LIST OF FIGURES 2.2 Diagram of reconstruction problems and results . ............. 77 2.3Cocircuitgraphofanorientedmatroidofrank2.............. 79 3.1 Diagrams of oriented matroids of rank 2 with 2 and 3 elements . 98 3.2 Diagrams of oriented matroids of rank 2 with 4 non-loop elements . 98 3.3 Relation of isomorphism classes of oriented matroids under single ele- ment extensions for n 5 .........................101 ≤ 3.4 Extension of pseudosphere arrangement . .............103 3.5Localizationoftopegraph.........................103 3.6Localizationofcocircuitgraph.......................104 5.1 The three possible types of a coline cycle . .............122 5.2 Infeasable assignment of a coline cycle . .............126 7.1 Sign vector defined by a hyperplane in a point configuration . 148 7.2 Point configuration and sphere arrangement . .............149 7.3 The order types with 3 and 4 non-collinear points in R2 ..........152 7.4 The 11 order types with 5 non-collinear points in R2; only the first 3 are non-degenerate . ...........................152 7.5 The 93 order types with 6 non-collinear points in R2; only the first 16 are non-degenerate . ...........................153 7.6 The counter-example with 9 points to Conjecture 7.5.1 . .........159 8.1 Hyperplane arrangement and sphere arrangement .............163 8.2 The dissection types with 2 and 3 non-parallel hyperplanes in R2 .....166 8.3 The 8 dissection types with 4 non-parallel hyperplanes in R2; only the first is non-degenerate . ...........................166 8.4 The 46 dissection types with 5 non-parallel hyperplanes in R2; only the first 6 are non-degenerate . ....................167 List of Tables 0.1 List of covectors in F .A/ .......................... 16 0.2 Faces and corresponding sign vectors ................... 19 6.1 Lexicographic and reverse lexicographic order of bases (n 5, r 3) . 135 = = 6.2 The 17 isomorphism classes in IC.6; 3/ ..................136 6.3Numberofisomorphismclassesoforientedmatroids...........141 6.4Numberofisomorphismclassesofuniformorientedmatroids.......141 6.5 Number of non-realizable isomorphism classes of uniform oriented matroids142 6.6 CPU time needed to compute isomorphism classes of oriented matroids . 143 6.7 Average CPU time needed to compute one isomorphism class (in mil- liseconds) . ..............................143 6.8 Memory used to store isomorphism classes of oriented matroids (in bytes, whereeverybytehas8bits)........................144 6.9 The 4 isomorphism classes in IC.5; 3/ ...................145 6.10 The 12 isomorphism classes in IC.6; 4/ ..................145 7.1 Number of abstract order types .......................151 7.2 Number of non-degenerate abstract order types . ............151 7.3 Number of non-realizable non-degenerate abstract order types of rank 3 . 151 7.4Numberofrelabelingclassesofmatroidpolytopes............154 7.5Numberofcombinatorialpolytopetypesofmatroidpolytopes......155 7.6 Number of combinatorial polytope types of simplicial matroid polytopes . 156 7.7CocircuitsoforientedmatroidviolatingConjecture7.5.2.........158 7.8CoordinatesofpointconfigurationviolatingConjecture7.5.1.......159 x LIST OF TABLES 7.9 Non-trivial collinearities of point configuration violating Conjecture 7.5.1 160 8.1 Number of abstract dissection types ....................165 8.2 Number of non-degenerate abstract dissection types . .........165 List of Pseudo-Codes 1.1 Algorithm ACYCLOIDORIENTATIONRECONSTRUCTION ......... 67 1.2 Algorithm LOWERFACES .......................... 70 1.3 Algorithm FACEENUMERATION ...................... 70 1.4 Algorithm COVECTORSFROMCOCIRCUITS ................ 72 1.5 Algorithm TOPESFROMCOCIRCUITS ................... 73 2.1 Algorithm OMLABELFROMMLABEL ................... 81 2.2 Input and Output Specification of LISTCOLINECYCLES ......... 83 2.3 Input and Output Specification of MLABELFROMCOLINECYCLES .... 83 2.4 Algorithm MLABELFROMAPLABEL ................... 84 4.1 Algorithm WEAKACYCLOIDALSIGNATURESREVERSESEARCH .....112 4.2 Algorithm WEAKACYCLOIDALSIGNATURESUPTOISOMORPHISM . 114 5.1 Algorithm LOCALIZATIONSPATTERNBACKTRACK ............129 Acknowledgements I am very grateful