Le théorème de Carathéodory coloré, prouvé en 1982 par Bárány, énonce le résultat suivant. Etant donnés d Å1 ensembles de points S1,SdÅ1 dans Rd , si chaque Si contient 0 dans son enveloppe convexe, alors il existe un sous-ensemble arc-en-ciel T µ SdÅ1 iÆ1 Si contenant 0 dans son enveloppe convexe, i.e. un sous-ensemble T tel que jT \Si j • 1 pour tout i et tel que 0 2 conv(T ). Ce théorème a donné naissance à de nombreuses questions, certaines algorithmiques et d’autres plus combinatoires. Dans ce manuscrit, nous nous intéressons à ces deux aspects. En 1997, Bárány et Onn ont défini la programmation linéaire colorée comme l’ensemble des questions algorithmiques liées au théorème de Carathéodory coloré. Parmi ces questions, deux ont particulièrement retenu notre attention. La première concerne la complexité du calcul d’un sous-ensemble arc-en-ciel comme dans l’énoncé du théorème. La seconde, en un sens plus générale, concerne la complexité du problème de décision suivant. Etant donnés des ensembles de points dans Rd , correspondant aux couleurs, il s’agit de décider s’il existe un sous-ensemble arc-en-ciel contenant 0 dans son enveloppe convexe, et ce en dehors des conditions du théorème de Carathéodory coloré. L’objectif de cette thèse est de mieux délimiter les cas polynomiaux et les cas “difficiles” de la programmation linéaire colorée. Nous présentons de nouveaux résultats de complexités permettant effectivement de réduire l’ensemble des cas encore incertains. En particulier, des versions combinatoires du théorème de Carathéodory coloré sont présentées d’un point de vue algorithmique. D’autre part, nous montrons que le problème de calcul d’un équilibre de Nash dans un jeu bimatriciel peut être réduit polynomialement à la programmation linéaire coloré. En prouvant ce dernier résultat, nous montrons aussi comment l’appartenance des problèmes de complémentarité à la classe PPAD peut être obtenue à l’aide du lemme de Sperner. Enfin, nous proposons une variante de l’algorithme de Bárány et Onn, calculant un sous ensemble arc-en-ciel contenant 0 dans son enveloppe convexe sous les conditions du théorème de Carathéodory coloré. Notre algorithme est clairement relié à l’algorithme du simplexe. Après une légère modification, il coïncide également avec l’algorithme de Lemke, calculant un équilibre de Nash dans un jeu bimatriciel. La question combinatoire posée par le théorème de Carathéodory coloré concerne le nombre de sous-ensemble arc-en-ciel contenant 0 dans leurs enveloppes convexes. Deza, Huang, Stephen et Terlaky (Colourful simplicial depth, Discrete Comput. Geom., 35, 597–604 (2006)) ont formulé la conjecture suivante. Si jSi j Æ d Å1 pour tout i 2 {1, . . . ,d Å1}, alors il y a au moins d2Å1 sous-ensemble arc-en-ciel contenant 0 dans leurs enveloppes convexes. Nous prouvons cette conjecture à l’aide d’objets combinatoires, connus sous le nom de systèmes octaédriques, dont nous présentons une étude plus approfondie / The colorful Carathéodory theorem, proved by Bárány in 1982, states the following. Given d Å1 sets of points S1, . . . ,SdÅ1 µ Rd , each of them containing 0 in its convex hull, there exists a colorful set T containing 0 in its convex hull, i.e. a set T µ SdÅ1 iÆ1 Si such that jT \Si j • 1 for all i and such that 0 2 conv(T ). This result gave birth to several questions, some algorithmic and some more combinatorial. This thesis provides answers on both aspects. The algorithmic questions raised by the colorful Carathéodory theorem concern, among other things, the complexity of finding a colorful set under the condition of the theorem, and more generally of deciding whether there exists such a colorful set when the condition is not satisfied. In 1997, Bárány and Onn defined colorful linear programming as algorithmic questions related to the colorful Carathéodory theorem. The two questions we just mentioned come under colorful linear programming. This thesis aims at determining which are the polynomial cases of colorful linear programming and which are the harder ones. New complexity results are obtained, refining the sets of undetermined cases. In particular, we discuss some combinatorial versions of the colorful Carathéodory theorem from an algorithmic point of view. Furthermore, we show that computing a Nash equilibrium in a bimatrix game is polynomially reducible to a colorful linear programming problem. On our track, we found a new way to prove that a complementarity problem belongs to the PPAD class with the help of Sperner’s lemma. Finally, we present a variant of the “Bárány-Onn” algorithm, which is an algorithmcomputing a colorful set T containing 0 in its convex hull whose existence is ensured by the colorful Carathéodory theorem. Our algorithm makes a clear connection with the simplex algorithm. After a slight modification, it also coincides with the Lemke method, which computes a Nash equilibriumin a bimatrix game. The combinatorial question raised by the colorful Carathéodory theorem concerns the number of positively dependent colorful sets. Deza, Huang, Stephen, and Terlaky (Colourful simplicial depth, Discrete Comput. Geom., 35, 597–604 (2006)) conjectured that, when jSi j Æ d Å1 for all i 2 {1, . . . ,d Å1}, there are always at least d2Å1 colourful sets containing 0 in their convex hulls. We prove this conjecture with the help of combinatorial objects, known as the octahedral systems. Moreover, we provide a thorough study of these objects

<p>This thesis deals with combinatorial and geometric aspects of linear optimization, and consists of two parts.</p> <p>In the first part, we address a conjecture formulated in 2008 and stating that the largest possible average diameter of a bounded cell of a simple hyperplane arrangement of n hyperplanes in dimension d is not greater than the dimension d. The average diameter is the sum of the diameters of each bounded cell divided by the total number of bounded cells, and then we consider the largest possible average diameter over all simple hyperplane arrangements. This quantity can be considered as an indication of the average complexity of simplex methods for linear optimization. Previous results in dimensions 2 and 3 suggested that a specific type of extensions, namely the covering extensions, of the cyclic arrangement might achieve the largest average diameter. We introduce a method for enumerating the covering extensions of an arrangement, and show that covering extensions of the cyclic arrangement are not always among the ones achieving the largest diameter.</p> <p>The software tool we have developed for oriented matroids computation is used to exhibit a counterexample to the hypothesized minimum number of external facets of a simple arrangement of n hyperplanes in dimension d; i.e. facets belonging to exactly one bounded cell of a simple arrangement. We determine the largest possible average diameter, and verify the conjectured upper bound, in dimensions 3 and 4 for arrangements defined by no more than 8 hyperplanes via the associated uniform oriented matroids formulation. In addition, these new results substantiate the hypothesis that the largest average diameter is achieved by an arrangement minimizing the number of external facets.</p> <p>The second part focuses on the colourful simplicial depth, i.e. the number of colourful simplices in a colourful point configuration. This question is closely related to the colourful linear programming problem. We show that any point in the convex hull of each of (d+1) sets of (d+1) points in general position in R^d is contained in at least (d+1)²/2 simplices with one vertex from each set. This improves the previously established lower bounds for d>=4 due to Barany in 1982, Deza et al in 2006, Barany and Matousek in 2007, and Stephen and Thomas in 2008.</p> <p>We also introduce the notion of octahedral system as a combinatorial generalization of the set of colourful simplices. Configurations of low colourful simplicial depth correspond to systems with small cardinalities. This construction is used to find lower bounds computationally for the minimum colourful simplicial depth of a configuration, and, for a relaxed version of the colourful depth, to provide a simple proof of minimality.</p> / Doctor of Philosophy (PhD)