Sur les aspects théoriques et pratiques des compromis dans les problèmes d'allocation des ressources. (On theoretical and practical aspects of trade-offs in resource allocation problems)

This thesis is divided into two parts. The first part of the thesis deals with the study of heuristic based approaches for approximating Pareto fronts. We propose a new Double Archive Pareto local search algorithm for solving multi-objective combinatorial optimization problems. We embed our technique into a genetic framework where our algorithm restarts with the set of new solutions formed by the recombination and the mutation of solutions found in the previous run. This method improves upon the existing Pareto local search algorithm for multiple instances of bi-objective and tri-objective quadratic assignment problem. In the second part of the thesis, we focus on non-preemptive scheduling algorithms. Here, we study the online problem of minimizing maximum stretch on a single machine. We present both positive and negative theoretical results. Then, we provide an optimally competitive semi-online algorithm. Furthermore, we study the problem of minimizing average stretch on a single machine in a recently proposed rejection model. We show that there exists a polynomial time O(1)-approximation algorithm for minimizing the average stretch and average flow time on a single machine. Essentially, our algorithm converts a preemptive schedule into a non-preemptive schedule by rejecting a small fraction of jobs such that their total weights are bounded. Lastly, we study the weighted average flow time minimization problem in online settings. We present a mathematical programming based framework that unifies multiple resource augmentations. Then, we developed a scheduling algorithm based on the concept of duality. We showed that our algorithm is O(1)-competitive for solving the weighted average flow time problem on a set of unrelated machines. Furthermore, we showed that our algorithm when applied to the problem of minimizing `k norms of weighted flow problem on a set of unrelated machines, is O(k)-competitive.