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Interactions Between Cliques and Stable Sets in a Graph Aurelie Lagoutte

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Interactions Between Cliques and Stable Sets in a Graph Aurelie Lagoutte Interactions between Cliques and Stable Sets in a Graph Aurelie Lagoutte To cite this version: Aurelie Lagoutte. Interactions between Cliques and Stable Sets in a Graph. Other [cs.OH]. Ecole normale supérieure de lyon - ENS LYON, 2015. English. NNT : 2015ENSL1012. tel-01208076 HAL Id: tel-01208076 https://tel.archives-ouvertes.fr/tel-01208076 Submitted on 1 Oct 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. -ÉCOLE NORMALE SUPÉRIEURE DE LYON - Laboratoire de l’Informatique du Parallélisme (LIP) - UMR 5668 THÈSE en vue d’obtenir le grade de Docteur de l’Université de Lyon, délivré par l’École Normale Supérieure de Lyon Discipline : Informatique au titre de l’École Doctorale Informatique et Mathématiques présentée et soutenue publiquement le 23 Septembre 2015 par Aurélie LAGOUTTE INTERACTIONS ENTRE LES CLIQUES ET LES STABLES DANS UN GRAPHE Directeur de thèse : Stéphan THOMASSÉ Devant la commission d’examen formée de : Samuel FIORINI Rapporteur Frédéric MAFFRAY Rapporteur Claire MATHIEU Examinatrice Mickaël MONTASSIER Examinateur Stéphan THOMASSÉ Directeur de thèse Contents Introduction (in French) 1 Introduction 11 1 Perfect graphs and Coloring 21 1.1 ContextandMotivations. ..... ...... ..... ...... .... 21 1.1.1 Perfectgraphs............................. 21 1.1.2 χ-boundedness ............................ 33 1.2 Excluding even holes except C4 ....................... 36 1.2.1 Forbidding5-holes . 38 1.2.2 Dealingwith5-holes . 43 2 The Erd˝os-Hajnal property 47 2.1 ContextandMotivations. ..... ...... ..... ...... .... 47 2.2 Strong Erd˝os-Hajnal property in (Pk, Pk)-freegraphs. 52 2.3 Usefultools .................................. 54 2.3.1 Unbalancing the Strong Erd˝os-Hajnal property . ..... 55 2.3.2 Easy neighborhood for the Erd˝os-Hajnal property . ..... 56 3 Clique-Stable Set Separation 59 3.1 DefinitionandContext ............................ 60 3.2 Randomgraphs ................................ 66 3.3 Forbiddingafixedsplitgraph. 69 3.4 Techniques in common with the Erd˝os-Hajnal property . ....... 75 3.4.1 (Pk, Pk)-freegraphs.......................... 76 3.4.2 The case of P5-freegraphs ...................... 77 3.4.3 Easyneighborhood..... ...... ..... ...... .... 78 3.4.4 Modules................................ 80 3.5 Perfectgraphswithnobalancedskew-partition . ...... 81 3.5.1 Definitions............................... 83 3.5.2 Decomposing trigraphs of .................... 87 F iv CONTENTS |v 3.5.3 Clique-StableSetseparation. 91 3.5.4 StrongErd˝os-Hajnalproperty. 97 4 Extended formulations 115 4.1 LinearProgramming ............................. 116 4.2 Polytopes.................................... 120 4.3 AlgorithmsforLP............................... 122 4.4 Totalunimodularity:whenIPmeetsLP . 123 4.5 STAB(G)inperfectgraphs . 125 4.6 ExtendedFormulation .. ...... ..... ...... ..... .... 130 4.7 TheSlackMatrixandtheFactorizationTheorem . 133 4.8 Linkwithcommunicationcomplexity . 137 4.9 ThebirthofCS-Separation. 139 5 Alon-Saks-Seymour conjecture 141 5.1 Fromoriginstodisproval . 142 5.2 Anorientedversion&equiv. with theCS-Separation . 144 5.3 Generalization to t-bicliquecovering . 148 6 Constraint Satisfaction Problems 151 6.1 Context..................................... 152 6.2 StubbornProblemandlinkwiththeCS-Separation . 154 Conclusion 163 vi| CONTENTS Remerciements Cette thèse n’aurait pas pu voir le jour sans un grand nombre de personnes qui m’en- tourent. C’est maintenant l’occasion de leur dédier ces quelques pages. Je tiens à remercier en tout premier lieu les membres de mon jury de m’avoir fait l’honneur d’évaluer cette thèse : merci à mes rapporteurs Frédéric MAFFRAY et Samuel FIORINI d’avoir relu avec autant d’attention mon manuscrit ; merci à Claire MATHIEU et Mickaël MONTASSIER d’avoir apporté leur expertise en ce jour de ma soutenance. Je ne sais que dire, Stéphan, pour te remercier d’avoir été mon direc- teur de thèse pendant ces 3 ans et d’avoir partagé ton immense savoir avec moi. Je resterai toujours impressionnée par ton incroyable intuition, qui te permet de savoir dans quelle direction aller, et par ta mémoire infaillible pour les références biblio- graphiques. Merci beaucoup pour ces discussions scientifiques qui m’ont permis de mener à bien cette thèse, même s’il a fallu quelques fois faire preuve de patience pour m’expliquer plusieurs fois la même chose. Un grand merci également de m’avoir soutenue et rassurée dans les moments de doute, et d’avoir toujours été disponible lorsque je venais toquer à ta porte. Merci enfin de m’avoir permis, par ton influence, de me rendre aux quatre coins du monde pour rencontrer les plus grands chercheurs de notre domaine. Pour avoir fait du labo un endroit où l’on se sent presque comme chez soi, je remercie tous mes collègues du LIP ; c’est à la fois effrayant et rassurant d’être en- tourée de si brillants chercheurs, et c’est d’autant plus stimulant qu’il y règne une excellente ambiance de travail. Un merci tout particulier aux membres passés et pré- sents de l’équipe MC2 pour leur accueil chaleureux et bien sûr pour les goûters. J’ai eu la chance de partager mon bureau avec plusieurs d’entre vous : merci à Bruno d’avoir eu réponse à toutes mes questions sur le monde de la recherche au tout début de ma thèse; merci à Sébastien, Théophile et Émilie (même si l’on n’a jamais offi- ciellement partagé un bureau) pour les quizzs musicaux ; merci à Sebastiàn et Mat- thieu, et désolée de vous avoir abandonnés si vite; et enfin merci à Ararat, Nick et Nacho pour la formidable bonne humeur qui a régné dans notre bureau ces huit derniers mois. Je garderai un souvenir impérissable de ces improbables cours de français/anglais/espagnol/arménien, et de notre grand complicité. Merci également à Natacha pour les nombreuses discussions diverses et variées, pour les échanges vii viii| REMERCIEMENTS de recettes et pour les comparatifs zumba/step/7 Minutes Workout (bizzarement, les autres membres de l’équipe étaient beaucoup moins sensibles à ces probléma- tiques). Merci à Nicolas d’avoir répondu à toutes mes questions sur les graphes par- faits mais aussi pour les blind tests Nostalgie lors des trajets en voiture. Je n’ou- blie pas, bien sûr, l’équipe des assistant(e)s sans qui la vie entière du labo se se- rait déjà écroulée. En particulier, merci à Marie, Chiraz et Damien d’avoir toujours su répondre à mes questions, et de vous être occupés de tous les papiers de mis- sion/réinscription/soutenance/(ceux dont je ne connais pas l’existence), et toujours avec le sourire. Merci aux doctorants des autres équipes avec qui j’ai eu l’occasion d’échanger (scientifiquement ou non), notamment lors du tout nouveau Séminaire des Docto- rants, ou bien autour d’un verre au foyer; à ce titre, je remercie tout particulièrement Gopi et Jean-Marie. Merci à Laure pour les discussions au détour d’un couloir, et pour avoir essayé de me faire faire de l’Archi. Je remercie également les collègues avec qui j’ai eu la chance d’enseigner ici à l’ENS de Lyon : Yves, Adeline, Pascal, Petru (merci pour les Kinder avant les TD de 8h), Maxime, Antoine, Omar et Aurélien ; et Damien et Éric, les grands chefs. L’enseignement est vraiment une facette du métier qui me tient à coeur, et j’en profite pour remercier mes étudiants de rendre cette tâche tou- jours aussi passionnante. J’ai eu la chance de me rendre régulièrement à des conférences, workshops ou autres Écoles de Recherche, et d’y rencontrer de nombreux chercheurs venant de diffé- rents horizons. Je remercie tout ceux que j’ai croisé, pour les discussions scientifiques, pour les discussions non-scientifiques, et pour cet incroyable sentiment d’apparte- nance à une communauté de recherche créative, stimulante et humaine. Je remercie en particulier les membres de STINT, pour les séminaires de travail réguliers et pour les moments de détente.Merci à Fred d’avoir relu une bonne moitié de mon manuscrit de son plein gré (est-ce pour me faire oublier ta tromperie sur le tour de VTT?). Merci à Nicolas, mon grand frère dans l’arbre généalogique des thèses, et merci à Marthe pour la complicité qui s’est installée au fil des ans. Je remercie, outre Stéphan qui a déjà eu son lot de louanges, Nicolas, Sébastien, Théophile, Aline, Zhentao, Petru et Julien avec qui j’ai eu la chance de collaborer pendant ma thèse (oui, certaines personnes apparaissent plusieurs fois dans ces Re- merciements). Je remercie également ceux avec qui j’ai partagé une excellente expé- rience de recherche pendant mes stages, bien avant de commencer ma thèse : merci à Andreas, Daniel et Fabienne, de l’Institut du Traitement des Langues Naturelles à Stuttgart; et bien sûr, merci à Mathilde et Éric, qui m’ont véritablement fait découvrir ce qu’était la Recherche en Informatique lors de mon tout premier stage en L3. Éric, je ne te remercierai jamais assez d’avoir été le premier à me convaincre de faire une thèse, parce que "finalement, c’est un peu comme le stage". Puisque je remonte le temps jusqu’à la L3, j’en profite pour remercier Paul GASTIN qui, à mon entrée à Cachan sur le concours info, ne m’as pas-vraiment-forcé-mais- quand-même-très-fortement-conseillé de ne pas bifurquer en maths, et de prendre le temps de découvrir ce qu’était véritablement l’Informatique Fondamentale. Je remer- cie les professeurs d’Informatique et de Mathématiques qui ont su me transmettre REMERCIEMENTS | ix leur passion pour la matière en elle-même et pour l’enseignement : en particulier, merci à Hubert COMON pour les « interrogations de questions », merci à Stéphane GONNORD pour avoir transformé chaque cours en véritable pièce de théâtre, et merci à M.
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