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## Section: New Results

### Online Non-preemptive Scheduling in a Resource Augmentation Model based on Duality

Resource augmentation is a well-established model for analyzing algorithms, particularly in the online setting. It has been successfully used for providing theoretical evidence for several heuristics in scheduling with good performance in practice. According to this model, the algorithm is applied to a more powerful environment than that of the adversary. Several types of resource augmentation for scheduling problems have been proposed up to now, including speed augmentation, machine augmentation and more recently rejection. In this paper , we present a framework that unifies the various types of resource augmentation. Moreover, it allows generalize the notion of resource augmentation for other types of resources. Our framework is based on mathematical programming and it consists of extending the domain of feasible solutions for the algorithm with respect to the domain of the adversary. This, in turn allows the natural concept of duality for mathematical programming to be used as a tool for the analysis of the algorithm’s performance. As an illustration of the above ideas, we apply this framework and we propose a primal-dual algorithm for the online scheduling problem of minimizing the total weighted flow time of jobs on unrelated machines when the preemption of jobs is not allowed. This is a well representative problem for which no online algorithm with performance guarantee is known. Specifically, a strong lower bound of $Ω\left(\surd n\right)$ exists even for the offline unweighted version of the problem on a single machine. In this paper, we first show a strong negative result even when speed augmentation is used in the online setting. Then, using the generalized framework for resource augmentation and by combining speed augmentation and rejection, we present an $\left(1+{ϵ}_{s}\right)$-speed $O\left(\frac{1}{{ϵ}_{s}{ϵ}_{r}}\right)$-competitive algorithm if we are allowed to reject jobs whose total weight is an ${ϵ}_{r}$-fraction of the weights of all jobs, for any ${ϵ}_{s}$ > 0 and ${ϵ}_{r}\in \left(0,1\right)$. Furthermore, we extend the idea for analysis of the above problem and we propose an $\left(1+{ϵ}_{s}\right)$-speed ${ϵ}_{r}$-rejection $O\left(\frac{{k}^{\frac{\left(k+3}{k}}}{{ϵ}_{r}^{1/k}{ϵ}_{s}^{\frac{\left(k+2\right)}{k}}}\right)$-competitive algorithm for the more general objective of minimizing the weighted ${l}_{k}$-norm of the flow times of jobs.