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).

In this paper, we deal with the problem of finding quasi-independent sets in graphs. This problem is formally defined in three versions, which are shown to be polynomially equivalent. The one that looks most general, namely, f -QIS, consists of, given a graph and a non-decreasing function f , finding a maximum size subset Q of the vertices of the graph, such that the number of edges in the induced subgraph is less than or equal to f (|Q|). For this problem, we show an exact solution method that runsn ) on graphs of average degree bounded by d. For the most specifically defined γ-QIS and k-QIS problems, several results on complexity and approximation are shown, and greedy algorithms are proposed, analyzed and tested.

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