Approximately: Independence Implies Vertex Cover

$\newcommand{\eps}{\varepsilon}$ We observe that a (1-\eps)-approximation algorithm to Independent Set, that works for any induced subgraph of the input graph, can be used, via a polynomial time reduction, to provide a (1+\eps)-approximation to Vertex Cover. This basic observation was made before, see [BHR11]. As a consequence, we get a PTAS for VC for unweighted pseudo-disks, QQPTAS for VC for unweighted axis-aligned rectangles in the plane, and QPTAS for MWVC for weighted polygons in the plane. To the best of our knowledge all these results are new

Le problème du prix de rang : une approche d'optimisation linéaire en nombres entiers mixtes

This doctorate is entirely devoted to an in-depth study of the Rank Pricing Problem (RPP) and two generalizations. The RPP is a combinatorial optimization problem which aims at setting the prices of a series of products of a company to maximize its revenue. This problem is specified by a set of unit-demand customers, that is, customers interested in a subset of the products offered by the company which intend to buy at most one of them. To do so, they count on a fixed budget, and they rank the products of their interest from the “best” to the “worst”. Once the prices are established by the company, they will purchase their highest-ranked product among the ones they can afford. In the RPP, it is assumed an unlimited supply of products, which is consistent with a company having enough copies of a product to satisfy the demand, or with a setting where the products can be produced quickly at negligible cost (e.g., digital goods). This dissertation consists of four chapters. The first chapter introduces the RPP problem and the mathematical concepts present in the work, whereas each of the next three chapters tackles the resolution of each of the problems of study: the RPP and two generalizations. Thus, Chapter 3 is dedicated to the Rank Pricing Problem with Ties (RPPT), an extension of the RPP where we consider that customers can express indifference among products in their preference list. And the last chapter of the thesis is devoted to a generalization of the problem that we have named the Capacitated Rank Pricing Problem (CRPP) with envy. For this generalization, we have considered reservation prices of customers for the different products that reflect their willingness to pay, instead of a single budget per customer. However, the main difference is that, in the CRPP, the company has a limited supply of products and might not be able to satisfy all the customers’ requests. This is a realistic assumption that we can find in many companies.The aim of this thesis is the proposal of mixed-integer linear formulations for the three problems of study, and their theoretical and/or computational comparison. The methodology used is based on the introduction of decision variables and adequate restrictions to model the problems. Another objective consists in strengthening the formulations by means of valid inequalities that reduce the feasible region of the relaxed problem and allow us to obtain better linear relaxation bounds. Finally, a third goal is to derive resolution algorithms for each of these models and compare them computationally, using commercial solvers.Cette thèse est consacrée à une étude approfondie du Rank Pricing Problem (RPP) et de deux généralisations. Le RPP est un problème d'optimisation combinatoire qui vise à fixer le prix des produits d'une entreprise afin de maximiser son profit. Elle concerne les clients à la demande, c'est-à-dire les clients qui sont intéressés par plusieurs produits de l'entreprise, mais qui n'ont l'intention d'en acheter qu'un. Les clients disposent d'un budget fixe et classent les produits qui les intéressent du "meilleur" au "pire". Lorsque l'entreprise fixe les prix, chaque client achètera son produit préféré parmi ceux qu'il peut se permettre. Dans le RPP, nous supposons que les produits sont offerts en quantité illimitée, ce qui convient si l'on considère que l'entreprise a suffisamment de produits pour satisfaire la demande, ou lorsque les produits peuvent être fabriqués rapidement avec un coût négligeable (par exemple, les biens numériques).Cette thèse se compose de quatre chapitres. Le premier est un chapitre d'introduction au problème et aux concepts mathématiques présents dans la thèse, tandis que les trois chapitres suivants se concentrent sur chacun des problèmes étudiés : le RPP et deux généralisations. Ainsi, le troisième chapitre est consacré à l'étude du Rank Pricing Problem with Ties (RPPT). Dans cette extension du problème, nous supposons que les clients peuvent exprimer leur indifférence entre les produits qui les intéressent au moyen de liens dans leur liste de préférences. Enfin, le dernier chapitre de la thèse comprend l'étude du Capacitated Rank Pricing Problem (CRPP) avec envie. Dans cette extension, nous avons supposé des prix de réserve pour les clients qui reflètent ce qu'ils sont prêts à payer pour chaque produit, plutôt qu'un budget unique par consommateur. Cependant, la principale différence est que dans le cas du CRPP, l'entreprise dispose d'un nombre limité de produits et peut ne pas être en mesure de satisfaire la demande de tous les clients. L'objectif de la thèse est d'obtenir des formulations linéaires en nombres entiers mixtes pour les trois problèmes étudiés, et de les comparer sur le plan théorique et/ou computationnel. À cette fin, la méthodologie utilisée est basée sur la proposition de variables de décision et de contraintes appropriées qui modélisent le problème. L'objectif suivant est l'amélioration de ces formulations au moyen d'inégalités valides qui réduisent l’ensemble admissible de la relaxation du problème et permettent d'obtenir une meilleure borne de la relaxation linéaire. Et enfin, un troisième objectif est la proposition d'algorithmes de résolution pour chacun de ces modèles, et leur comparaison ultérieure au moyen d'études computationnelles et de résolution au moyen de logiciels d'optimisation commerciaux

Discrete Optimization in Early Vision - Model Tractability Versus Fidelity

Early vision is the process occurring before any semantic interpretation of an image takes place. Motion estimation, object segmentation and detection are all parts of early vision, but recognition is not. Some models in early vision are easy to perform inference with---they are tractable. Others describe the reality well---they have high fidelity. This thesis improves the tractability-fidelity trade-off of the current state of the art by introducing new discrete methods for image segmentation and other problems of early vision. The first part studies pseudo-boolean optimization, both from a theoretical perspective as well as a practical one by introducing new algorithms. The main result is the generalization of the roof duality concept to polynomials of higher degree than two. Another focus is parallelization; discrete optimization methods for multi-core processors, computer clusters, and graphical processing units are presented. Remaining in an image segmentation context, the second part studies parametric problems where a set of model parameters and a segmentation are estimated simultaneously. For a small number of parameters these problems can still be optimally solved. One application is an optimal method for solving the two-phase Mumford-Shah functional. The third part shifts the focus to curvature regularization---where the commonly used length and area penalization is replaced by curvature in two and three dimensions. These problems can be discretized over a mesh and special attention is given to the mesh geometry. Specifically, hexagonal meshes in the plane are compared to square ones and a method for generating adaptive meshes is introduced and evaluated. The framework is then extended to curvature regularization of surfaces. Finally, the thesis is concluded by three applications to early vision problems: cardiac MRI segmentation, image registration, and cell classification

Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)

The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum