Finding a subdivision of a digraph

International audienceWe consider the following problem for oriented graphs and digraphs: Given a directed graph D, does it contain a subdivision of a prescribed digraph F? We give a number of examples of polynomial instances, several NP-completeness proofs as well as a number of conjectures and open problems

Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes

Implementation of Bourbaki's Elements of Mathematics in Coq: Part Two; Ordered Sets, Cardinals, Integers

We believe that it is possible to put the whole work of Bourbaki into a computer. One of the objectives of the Gaia project concerns homological algebra (theory as well as algorithms); in a first step we want to implement all nine chapters of the book Algebra. But this requires a theory of sets (with axiom of choice, etc.) more powerful than what is provided by Ensembles; we have chosen the work of Carlos Simpson as basis. This reports lists and comments all definitions and theorems of the Chapter ''Ordered Sets, Cardinals, Integers''. Version 3 is based on the Coq ssreflect library. Version 5 implements many properties of ordinal numbers and infinite cardinal numbers. Version 6 includes the Veblen hierarchy of ordinals, the Schütte function psi, and a bit of theory of models.Version 7 includes rational and real numbers. Versions 8 and 9 include more theorems about ordinal numbers. Version 9 includes Sperner's theorem, and corrects a mistake in the size of one. The code (including some exercises) is available on the Web, under http://www-sop.inria.fr/marelle/gaia .Nous pensons qu’il est possible de mettre dans un ordinateur l’ensemble de l’œuvre de Bourbaki. L’un des objectifs du projet Gaia concerne l’algèbre homologique (théorie et algorithmes); dans une première étape nous voulons implémenter les neuf chapitres du livre Algèbre. Au préalable, il faut implémenter la théorie des ensembles. Nous utilisons l’Assistant de Preuve Coq; les choix fondamentaux et axiomes sont ceux proposés par Carlos Simpson. Ce rapport liste et commente toutes les définitions et théorèmes du Chapitre “Ensembles ordonnés, cardinaux, nombres entiers”. La version 9 de ce document décrit la bibliothèque à la fin de l'année 2017. Une partie des exercises a été résolue. Le code est disponible sur le site Web http://www-sop.inria.fr/marelle/gai