On some problems related to 2-level polytopes Thèse n. 9025 2018 présenté le 22 Octobre 2018 à la Faculté des Sciences de Base chaire d'optimisation discrète programme doctoral en mathématiques École Polytechnique Fédérale de Lausanne pour l'obtention du grade de Docteur ès Sciences par Manuel Francesco Aprile acceptée sur proposition du jury: Prof Hess Bellwald Kathryn, président du jury Prof Eisenbrand Friedrich, directeur de thèse Prof Faenza Yuri, codirecteur de thèse Prof Svensson Ola, rapporteur Prof Fiorini Samuel, rapporteur Prof Weltge Stefan, rapporteur Lausanne, EPFL, 2018 "You must go from wish to wish. What you don’t wish for will always be beyond your reach." — Michael Ende, The Neverending Story. To my brother Stefano, to whom I dedicated my Bachelor thesis too. He is now six years old and he has not read it yet. Maybe one day? Acknowledgements My PhD has been an incredible journey of discovery and personal growth. There were mo- ments of frustration and doubt, when research just wouldn’t work, even depression, but there was also excitement, enjoyment, and satisfaction for my progresses. Oh, there was hard work too. A whole lot of people contributed to making this experience so amazing. First, I would like to thank my advisor Fritz Eisenbrand for his support, the trust he put in me, and for letting me do research in total freedom. He allowed me to go to quite some conferences and workshops, where I could expand my knowledge and professional network, as well as enjoying some traveling. A big “grazie” goes to Yuri Faenza, my co-advisor: in a moment when I was not sure about the direction of my research, he gave a talk about 2-level polytopes, immediately catching my interest, and, a few days after, he was introducing me to extended formulations with an improvised crash course. Ironically, that day he explained to me Yannakakis’ protocol, which became the core of the last chapter of my thesis. He invested many hours working with me, taught me so much, he even hosted me at his place in New York (thanks to his wife Fanny too!): it goes without saying that this thesis would not have been possible without him. I am indebted to all researchers I met, from which I gathered interesting conversations, inspira- tion, precious pieces of advice. A complete list would be too long, but I would like to mention Michele Conforti, Marco Di Summa, Gwenaël Joret, Volker Kaibel, Monaldo Mastrolilli, Janos Pach. In particular I thank Kathryn Hess Bellwald, Ola Svensson, Stefan Weltge for being part of my PhD jury. Heartfelt thanks to Samuel Fiorini for inviting me to Bruxelles and for teaching me so much in so little time. Thanks to Tony Huynh and Marco Macchia, it was great to work and go for beers with you. I look forward to repeating both experiences. Thanks to Alfonso Cevallos, best office mate ever, for all the adventures, discussions, and for solving problems that I could not solve. Thanks to each member of DISOPT group, in particular to Igor Malinovi´c and Christoph Hunkenschröder for their valuable friendship and all the fun activities we did together, and to the members of DCG group too. We had quite a good time! I am also very grateful to Jocelyne and Natacha for their kindness and precious administrative support. I also had the luck to make many friends in the math department and beyond, among which I would like to mention Riccardo (worst housemate!), Clelia, Francesco, Gennaro, Jessica, Arwa, Lydia, Thomas, Chiara, Mauro. They might not have directly contributed to my work, but definitely made my time off much more enjoyable. Finally, the support and the love I had from Sicily was very important to me, especially during difficult moments. Thanks to all my Sicilian friends. Thanks to Carla, who made the last part of my PhD way better than the first in so many aspects: trying to explain my research to you v Acknowledgements was so fun that I plan to do it again and again, you have been warned. I am much grateful to all my family and in particular to my parents, for being extremely supportive and for encouraging me to pursue my dreams with no limits but my ambition, even though this meant to go far from home. Grazie. Lausanne, 17 September 2018 vi Abstract In this thesis we investigate a number of problems related to 2-level polytopes, in particular regarding their combinatorial structure and extension complexity. 2-level polytopes have been introduced as a generalization of stable set polytopes of perfect graphs, and despite their apparently simple structure, are at the center of many open problems: these include connec- tion with communication complexity and the separation between linear and semidefinite programming. The extension complexity of a polytope P is a measure of the complexity of representing P: it is the smallest size of an extended formulation of P, which in turn is a linear description of a polyhedron that projects down to P. In the first chapter we introduce the main concepts that will be used through the thesis and we motivate our interest in 2-level polytopes. In the second chapter we examine several classes of 2-level polytopes arising in combinatorial settings and we prove a relation between the number of vertices and facets of such polytopes, which is conjectured to hold for all 2-level polytopes. The proofs are obtained through an improved understanding of the combinatorial structure of such polytopes, which in some cases leads to results of independent interest. In the third chapter we study the extension complexity of a restricted class of 2-level polytopes, the stable set polytopes of bipartite graphs, for which we obtain improved lower and upper bounds. In the fourth chapter we study slack matrices of 2-level polytopes, important combinatorial objects related to extension complexity, defining operations on them and giving algorithms for the following recognition problem: given a matrix, determine whether it is a slack matrix of some special class of 2-level polytopes. In the fifth chapter we address the problem of explicitly obtaining small size extended formu- lations whose existence is guaranteed by communication protocols. In particular we give an output-efficient algorithm to write down extended formulations for the stable set polytope of perfect graphs, making a well known result by Yannakakis constructive, and we extend this to all deterministic protocols. We then conclude the thesis outlining the main open questions that stem from our work. Keywords: Polytopes, polyhedral combinatorics, 2-level, extension complexity, vertices, facets, slack matrix. vii Sommario In questa tesi vengono trattati diversi problemi sui politopi "2-level", in particolare sulla loro struttura combinatoria e complessità di estensione. Tali politopi sono una generalizzazione di politopi che derivano dagli insiemi indipendenti nei grafi perfetti, e, nonostante la loro struttura apparentemente semplice, sono al centro di molti problemi aperti che spaziano dalla complessità computazionale alla programmazione semidefinita. La complessità di estensione di un politopo P è una misura della complessità nel rappresentare P: è la minima dimensione di una formulazione estesa di P, che a sua volta è una descrizione lineare di un poliedro di cui P è la proiezione. Nel primo capitolo vengono introdotti i politopi 2-level e i concetti principali che verran- no usati nella tesi, e vengono descritte le principali motivazioni dell’interesse verso questi politopi. Nel secondo capitolo vengono esaminate diverse classi di politopi 2-level che appaiono in contesti combinatori, e viene provata una relazione tra il numero di faccette e di vertici di tali politopi. Congetturiamo che tale relazione valga per tutti i politopi 2-level. Le dimostrazioni vengono ottenute tramite una migliore comprensione della struttura combinatoria di tali politopi, che a volte porta a risultati interessanti a prescindere dalla congettura. Nel terzo capitolo studiamo la complessità di estensione di una particolare classe di politopi 2-level, derivante dagli insiemi indipendenti dei grafi bipartiti, di cui miglioriamo il limite inferiore e superiore. Nel quarto capitolo studiamo le matrici di slack dei politopi 2-level, importanti oggetti combi- natori collegati alla complessità di estensione, definiamo operazioni su tali matrici e diamo algoritmi per il seguente problema: data una matrice, determinare se è una matrice di slack di una certa classe di politopi 2-level. Nel quinto capitolo affrontiamo il problema di ottenere formulazioni estese compatte ed espli- cite, quando l’esistenza di tali formulazioni è dimostrata tramite protocolli di comunicazione. In particolare diamo un algoritmo per ottenere una formulazione estesa del politopo degli insiemi indipendenti dei grafi perfetti, rendendo costruttivo un noto risultato di Yannakakis. Il risultato è abbastanza generale da essere applicabile a tutti i protocolli deterministici. La tesi si conclude con una discussione delle principali direzioni di ricerca che scaturiscono dal nostro lavoro. Parole chiave: politopi, 2-level, formulazioni estese, vertici, faccette, matrice di slack. ix Contents Acknowledgementsv Abstract (English/Italiano) vii 1 Introduction 1 1.1 Preliminaries.......................................6 2 On vertices and facets of 2-level polytopes arising in combinatorial settings7 2.1 Introduction........................................7 2.2 Basics............................................9 2.2.1 Hanner and Birkhoff polytopes.........................9 2.3 Graphical 2-Level Polytopes............................... 10 2.3.1 Stable set polytopes of perfect graphs..................... 11 2.3.2 Hansen polytopes................................. 12 2.3.3 Min up/down polytopes............................. 12 2.3.4 Polytopes coming from posets......................... 14 2.3.5 Stable matching polytopes........................... 16 2.4 2-Level Matroid Base Polytopes............................. 19 2.4.1 2-level matroid polytopes and Conjecture 2.1................ 20 2.4.2 Flacets of 2-sums................................. 24 2.4.3 Linear Description of 2-Level Matroid Base Polytopes..........