Projet de thÃ¨se en Informatique
Sous la direction de Claire Mathieu et de Jose Correa.
ThÃ¨ses en prÃ©paration Ã Paris Sciences et Lettres en cotutelle avec l'Universidad de Chile , dans le cadre de Ã‰cole doctorale de Sciences mathÃ©matiques de Paris Centre (Paris) , en partenariat avec LIENS - Laboratoire d'informatique de l'Ã‰cole normale supÃ©rieure (laboratoire) et de Ã‰cole normale supÃ©rieure (Paris ; 1985-....) (Ã©tablissement de prÃ©paration de la thÃ¨se) depuis le 01-10-2014 .
The main objective of this thesis is to study the power of LP/SDP hierarchies for some scheduling problems relaxations, and some graph optimization problems. The different research lines are the following: Â• The design of approximation algorithms. Just a few results for scheduling problems use these approach, and among them we can find an application for scheduling on 2 machines with precedence constraints due to Svensson, and lower bounds for the problem of minimizing the weighted sum of tardy jobs, due to Mastrolilli [19]. The first result corresponds to a positive result, in the sense that one round of the Lasserre Hierarchy over a time-indexed linear relaxation is enough to reach the integer hull. However, for the min-sum tardy jobs problem, Mastrolilli showed that at level Î©( n) the integrality gap remains unbounded. This problem has a a connection with the Min-Knapsack problem and Lasserre is weak in these cases, even when both problems admit FPTAS. Â• Gap separation between Lasserre and other hierarchies. For example, Li and Svensson [22] showed recently an approximation algorithm for the Uncapacitated Facility Location problem, inspired in the Sherali-Adams hierarchy. It is interesting to understand how the Lasserre hierarchy behaves on problems with this structure. Â• Strong relaxations. The Sum-of Squares hierarchy might be a powerful complement for already known strong relaxations, like configuration LP's, and it can be helpful to improve known approximation algorithms or to study the hardness of the problem.
Linear and semidefinite programming for scheduling and graph optimization problems
The main objective of this thesis is to study the power of LP/SDP hierarchies for some scheduling problems relaxations, and some graph optimization problems. The different research lines are the following: Â• The design of approximation algorithms. Just a few results for scheduling problems use these approach, and among them we can find an application for scheduling on 2 machines with precedence constraints due to Svensson, and lower bounds for the problem of minimizing the weighted sum of tardy jobs, due to Mastrolilli [19]. The first result corresponds to a positive result, in the sense that one round of the Lasserre Hierarchy over a time-indexed linear relaxation is enough to reach the integer hull. However, for the min-sum tardy jobs problem, Mastrolilli showed that at level Î©( n) the integrality gap remains unbounded. This problem has a a connection with the Min-Knapsack problem and Lasserre is weak in these cases, even when both problems admit FPTAS. Â• Gap separation between Lasserre and other hierarchies. For example, Li and Svensson [22] showed recently an approximation algorithm for the Uncapacitated Facility Location problem, inspired in the Sherali-Adams hierarchy. It is interesting to understand how the Lasserre hierarchy behaves on problems with this structure. Â• Strong relaxations. The Sum-of Squares hierarchy might be a powerful complement for already known strong relaxations, like configuration LP's, and it can be helpful to improve known approximation algorithms or to study the hardness of the problem.