Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs-a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a much richer class of graphs. In particular, mixed unit interval graphs are not claw-free, compared to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs. The original bubble model was used by Boyaci, Ekim, and Shalom for proving the polynomiality of the MaxCut problem on unit interval graphs. However, we found a significant mistake in the proof which seems to be hardly repairable. Moreover, we demonstrate advantages of such model by providing a subexponential-time algorithm solving the MaxCut problem on mixed unit interval graphs using our extended version of the bubble model. In addition, it gives us a polynomial-time...

The problem of extending partial representations for an interval graph asks, whether it is possible to extend a given representation of some vertices to a valid representation of the entire graph. In this thesis we extend the recent result of Klavík et al. who proved REPEXT can be decided for proper and unit interval graphs in polynomial time. We describe properties of PI± and U± graphs and their representations and present algorithms deciding REPEXT for these classes in polynomial time. In the process, we characterize relations between the K1,3's in a graph and show that we can decide the open vertex of every K1,3. We also define notions of representation of the same order type and locally similar representations as well as intervals forced and locally forced to be closed (open) that are essential for extending partial representations when multiple types of intervals can occur in the same representation. We characterize intervals forced and locally forced to be closed (open) in a U± graph using integer gaps in the pre-representation and we construct lower bounds for the rightmost endpoint of a component in polynomial time.

Le problème de placement sur deux dimensions consiste à décider s’il existe un rangement d’objets rectangulaires dans une boîte donnée. C’est un problème combinatoire difficile (à la complexité du respect des capacités s’ajoute celle du positionnement des objets).Dans cette thèse, nous considérons les variantes sans rotation des objets et avec ou sansoptimisation de la valeur des objects placés.Nous menons une étude exploratoire des méthodologies qui peuvent être développéesà l’interface de la programmation mathématique, de l’optimisation combinatoire et de lathéorie des graphes. Notre objectif est aussi de développer des approches non basées surune discrétisation de la boîte, les plus performantes à l’heure actuelle.Dans ce mémoire, nous effectuons d’abord une étude théorique des qualités de bornesqui peuvent être obtenues avec les différentes formulations classiques. Au cours de cetteétude, nous renforçons certaines de ces formulations et en proposons de nouvelles formulations. Une étude qualitative des bornes issues de la relaxation linéaire des formulationstestés sur des jeux d’instances classiques de la littérature confirme l’étude théorique. Cetteétude permet de se rendre compte des facteurs déterminant la qualité des bornes et desenjeux à relever par la programmation mathématique.Par la suite, nous avons développé et testé deux approches de résolution innovantes.L’une est basée sur la décomposition de Dantzig-Wolfe associée à un branchement surles contraintes disjonctives de non recouvrement des objets. Cette approche a permis uneamélioration des résultats obtenus par la programmation mathématique.L’autre approche constitue en une approche combinatoire basée sur diverses caractérisations des graphes d’intervalles (modélisant le chevauchement des objets selon leurprojection sur chaque axe). Un premier algorithme est basé sur l’énumération de matricesde uns-consécutifs. Un autre utilise des arbres étiquetés pour éliminer plus efficacement lescas de symétries entre placements. Ces approches ont l’avantage de ne pas dépendre d’unediscrétisation du conteneur / The two dimensional orthogonal packing problem consists in deciding whether thereexists a packing of rectangular items in a given bin. This is a hard combinatorial problem(in addition to capacity constraints, one has to face the complexity of item positionning).In this thesis, we consider the case without item rotation and with or without packingvalue optimization.We explore methodologies at the interface of mathematical programming, combinatorial optimization and graph theory. Our aim is also to develop approaches not based on abin discretization (i.e. an alternative to such methods that are currently the most effective).In this work, we perform a theorical study of the quality of bounds of differents classicalformulations. We tighten some formulations and we propose new formulations. We perform a numerical study to test bound quality on classical instances. This study permits toidentify the determinant factor in the quality of mathematical programming formulations.We develop and test two resolution approaches. The first is based on Dantzig-Wolfedecomposition associated with a branching on no-overlapping disjunctive constraints. Thisapproach permits to improve results obtained by mathematical programming.The second approach establish a combinatorial approach based on multiple intervalgraph caracterization (modelling the item no-overlapping according to their projection oneach axis). The first algorithm is based on consecutive ones matrices enumeration. An otheruse labelled tree to eliminate more efficiently symmetry in packing. These approaches haveto advantage of being independent from bin discretization