Les matroïdes et leur implication dans l'allocation de ressources indivisibles : algorithmes d'approximation avec garantie de performance

In this thesis, we are interested in collective decision-making. The objective is to find a tradeoff solution for problems that are evaluated by multiple points of view. We consider problems having a matroid structure. Matroid theory is significant in combinatorial optimization, it helped to unify apparently separated structures like forests and matchings in graphs and it includes efficient algorithms for solving non-trivial optimization problems in polynomial time. We are interested to provide polynomial time centralized and decentralized approximation algorithms for finding a tradeoff solution which is a base of the matroid. The tradeoff solution must also be fair for all the members of the community. We are particularly interested in the issue of the fair division of indivisible goods which is central in computational social choice and that can be modeled by matroids.