Optimization and Realizability Problems for Convex Geometries Thesis presented by Keno MERCKX In fulfillment of the requirements for the degree of Doctor of Philosophy (Docteur en Sciences) Academic year 2018-2019 Supervisor: Jean CARDINAL Co-supervisor: Jean-Paul DOIGNON Samuel FIORINI (Universite´ libre de Bruxelles, Chair) Gwenael¨ JORET (Universite´ libre de Bruxelles, Secretary) Stefan LANGERMAN (Universite´ libre de Bruxelles) Michel HABIB (Universite´ Paris Diderot) Maurice QUEYRANNE (University of British Columbia) i UNIVERSITE´ LIBRE DE BRUXELLES DOCTORAL THESIS Optimization and Realizability Problems for Convex Geometries Author: Supervisor: Keno MERCKX Jean CARDINAL Co-supervisor: Jean-Paul DOIGNON Algorithms Research Group Departement´ d’Informatique iii “Pluralitas non est ponenda sine necessitate.” William of Ockham v UNIVERSITE´ LIBRE DE BRUXELLES Faculte´ des Sciences Departement´ d’Informatique Abstract Optimization and Realizability Problems for Convex Geometries by Keno MERCKX Convex geometries are combinatorial structures; they capture in an ab- stract way the essential features of convexity in Euclidean space, graphs or posets for instance. A convex geometry consists of a finite ground set plus a collection of subsets, called the convex sets and satisfying certain axioms. In this work, we study two natural problems on convex geometries. First, we consider the maximum-weight convex set problem. After proving a hard- ness result for the problem, we study a special family of convex geometries built on split graphs. We show that the convex sets of such a convex geom- etry relate to poset convex geometries constructed from the split graph. We discuss a few consequences, obtaining a simple polynomial-time algorithm to solve the problem on split graphs. Next, we generalize those results and design the first polynomial-time algorithm for the maximum-weight convex set problem in chordal graphs. Second, we consider the realizability problem. We show that deciding if a given convex geometry (encoded by its copoints) results from a point set in the plane is R-hard. We complete our text with a 9 brief discussion of potential further work. vii Acknowledgements First and foremost, I would like to thank Jean Cardinal and Jean-Paul Doignon for having accepted to supervise this thesis, and for their constant support during these six years. I am also very grateful for the freedom you allowed me on how I chose to organize my work. Thank you for guiding me through this journey and for all the feedback on this document. During my time as a student at ULB, I had the chance to meet excellent teachers. They deeply changed my vision of mathematics and computer sci- ence. Obviously Jean Cardinal and Jean-Paul Doignon, but also Samuel Fior- ini with his lectures on approximation algorithms, Gwenael¨ Joret by sharing his love for graph theory, and Olivier Markowitch and Yves Roggeman with their lectures on cryptography. I am also deeply thankful to Stefan Langer- man for his computation geometry class, and without whom I could not have started this work. My gratitude goes to Michel Habib and Maurice Queyranne (in addition to Jean Cardinal, Jean-Paul Doignon, Samuel Fiorini, Stefan Langerman, and Gwenael¨ Joret who have already been cited), for accepting to be part of the jury and for reviewing my thesis. Many thanks are also due to my friend Franc¸ois, with whom it was al- ways a pleasure to “work” with and share our happiness and frustration. I would like to say thank you to Udo for his interest in my work and his help- ful and challenging collaboration. I am grateful to Jeremie, Julie, Thibaut, Matthieu, Patrick, Christine and Hoan-Phung for making this experience amazing. I would like to thank the colleagues who worked as partners in different teaching activities. I would also like to thank my former classmates Isabelle, Maxime, and Rachel from ULB, and Michael, Coralie and Antoine from ARW. A huge thanks to Lucien, Olivier, Denis, Loraine, Laura, Lionel, Eiman and Celine.´ It goes without saying that I am greatly indebted to my parents as well as to my sister for their continuous support, encouragement, and love. Last, but not least, a heartfelt thank you goes to Marlene` for all your love, support and patience when I was only thinking about strange drawings. I would like to finish the acknowledgments by thanking any people that I may have forgotten to mention for their support and advice throughout the years. ix Contents Abstractv Acknowledgements vii 1 Introduction: summary and main results1 1.1 Convex geometries and antimatroids...............1 1.2 Finding maximum-weight convex/feasible sets.........2 1.3 The realizability problem......................3 1.4 Contributions and collaborations.................4 2 Background5 2.1 Basics.................................5 2.2 Graph theory............................5 2.3 Order theory.............................6 2.4 Geometry...............................7 2.5 Complexity theory.........................7 3 An abstract notion of convexity9 3.1 Convex geometries.........................9 3.1.1 From closure operators to convex geometries......9 3.1.2 Antimatroids and shellings................ 12 3.1.3 Copoints and bases..................... 14 3.1.4 Free sets, circuits and roots................ 15 3.1.5 Occurrences, applications and research topics..... 17 3.2 Examples of convex geometries.................. 19 3.2.1 Convex geometries on posets............... 19 3.2.2 Shellings of chordal graphs................ 20 3.2.3 Affine convex geometries................. 22 3.2.4 Search antimatroids in (directed) graphs........ 24 3.2.5 Miscellaneous........................ 25 x 4 The maximum-weight convex set problem 27 4.1 A classic optimization problem.................. 27 4.1.1 Problem definition..................... 28 4.1.2 Computational hardness.................. 29 4.1.3 Note on polyhedral results................ 31 4.2 Special cases solvable in polynomial time............ 31 4.2.1 Result for poset convex geometries............ 31 4.2.2 Result for double poset convex geometries....... 33 4.2.3 Result for tree convex geometries on vertices...... 34 4.2.4 Result for tree convex geometries on edges....... 35 4.2.5 Result for affine convex geometries in the plane.... 35 4.3 The case of split graphs....................... 36 4.3.1 Characterization of the feasible sets........... 36 4.3.2 Connection between split graph shellings and posets. 41 4.3.3 The base poset....................... 44 4.3.4 Optimization results.................... 46 4.3.5 Free sets and circuits characterization.......... 47 4.3.6 Beyond this special case.................. 49 5 Finding a maximum-weight convex set in a chordal graph 51 5.1 More on chordal graphs...................... 51 5.1.1 Definitions.......................... 51 5.1.2 The clique-separator graph................ 52 5.2 Problems............................... 53 5.2.1 Main problem........................ 53 5.2.2 Dummy vertices and sub-problems........... 54 5.3 A special case solvable in polynomial time........... 55 5.3.1 The rooted poset...................... 56 5.3.2 Reduction to a poset problem............... 58 5.4 A polynomial-time algorithm................... 60 5.4.1 Computation phase..................... 61 5.4.2 Preprocessing........................ 63 5.5 Analysis............................... 64 5.5.1 Time complexity...................... 65 5.5.2 Detailed example...................... 67 xi 6 The realizability problem for convex geometries 71 6.1 Basics of computational geometry................ 71 6.1.1 Affine convex geometries................. 72 6.1.2 Abstract order types and chirotopes........... 73 6.1.3 Existential theory of the reals............... 76 6.2 Hardness result for the realizability problem.......... 77 6.2.1 Overview.......................... 77 6.2.2 Technical properties.................... 78 6.2.3 The fixed ring........................ 83 6.2.4 The reduction........................ 84 7 Conclusion 89 7.1 Further work............................. 89 7.2 Closing remark........................... 90 A Allowable sequences and realizability problems 91 A.1 Allowable sequences........................ 91 A.2 Realizability for allowable sequence............... 94 A.3 Results for simple allowable sequences............. 95 A.4 Realizability for convex geometries................ 96 Bibliography 101 Index 113 1 Chapter 1 Introduction: summary and main results “Well, it’s rather difficult to define. Perhaps I’m just projecting my own concern about it. I know I’ve never completely freed myself of the suspicion that there are some extremely odd things about this mission. I’m sure you’ll agree there’s some truth in what I say.” — HAL 9000 in 2001: A Space Odyssey We consider two natural problems on convex geometries: the maximum- weight convex set problem and the realizability problem. The main goal of this work is to obtain computational results for those questions on specific families of convex geometries. The theorems presented in this section are the main original results of this thesis. The notions used here will be formally introduced later on with references and detailed examples. 1.1 Convex geometries and antimatroids Chapter3 of this work presents the concept of convex geometries, together with some examples. Roughly said, a convex geometry is defined by a finite set, called the ground set, and some special subsets of this ground set that we call “convex sets”. These special subsets must share some well-defined properties. Formally, a set system (V, ) is a convex geometry if ? , the C 2 C set is stable under intersection, and for all C in V , there exists a c in C C n f g V C such that C c is also in . We also study structures related to convex n [ f g C geometries: antimatroids. Formally, a set system (V, ) is an antimatroid F if V , the set is stable under union, and for all F in ? , there 2 F F F n f g exists an f in F such that F f is also in . The sets in are called the n f g F F feasible sets. Convex geometries are related to antimatroids in the following 2 Chapter 1. Introduction: summary and main results sense: (V, ) is a convex geometry if and only if (V, V C : C ) is an C f n 2 Cg antimatroid.