We study an online model for the maximum k-coverage problem, where given a universe of elements E = {e 1 , e 2 ,. .. , e m }, a collection of subsets of E, S = {S 1 , S 2 ,. .. , S n }, and an integer k, we ask for a subcollection A ⊆ S, such that |A| = k and the number of elements of E covered by A is maximized. In our model, at each step i, a new set S i is revealed, and we have to decide whether we will keep it or discard it. At any time of the process, only k sets can be kept in memory; if at some point the current solution already contains k sets, any inclusion of any new set in the solution must entail the irremediable deletion of one set of the current solution (a set not kept when revealed is irremediably deleted). We first propose an algorithm that improves upon former results for the same model. We next settle a graph-version of the problem, called maximum k-vertex coverage problem. Here also we propose non-trivial improvements of the competitive ratio for natural classes of graphs (mainly regular and bipartite).

We propose a unifying framework based on configuration linear programs and randomized rounding, for different energy optimization problems in the dynamic speedscaling setting. We apply our framework to various scheduling and routing problems in heterogeneous computing and networking environments. We first consider the energy minimization problem of scheduling a set of jobs on a set of parallel speed scalable processors in a fully heterogeneous setting. For both the preemptive-non-migratory and the preemptive-migratory variants, our approach allows us to obtain solutions of almost the same quality as for the homogeneous environment. By exploiting the result for the preemptive-non-migratory variant, we are able to improve the best known approximation ratio for the single processor non-preemptive problem. Furthermore, we show that our approach allows to obtain a constant-factor approximation algorithm for the power-aware preemptive job shop scheduling problem. Finally, we consider the min-power routing problem where we are given a network modeled by an undirected graph and a set of uniform demands that have to be routed on integral routes from their sources to their destinations so that the energy consumption is minimized. We improve the best known approximation ratio for this problem.