Online maximum k-coverage

We study an online model for the maximum k-vertex-coverage problem, where given a graph G = (V,E) and an integer k, we ask for a subset A ⊆ V, such that |A| = k and the number of edges covered by A is maximized. In our model, at each step i, a new vertex vi is revealed, and we have to decide whether we will keep it or discard it. At any time of the process, only k vertices can be kept in memory; if at some point the current solution already contains k vertices, any inclusion of a new vertex in the solution must entail the definite deletion of another vertex of the current solution (a vertex not kept when revealed is definitely deleted). We propose algorithms for several natural classes of graphs (mainly regular and bipartite), improving on an easy 1/2-competitive ratio. We next settle a set-version of the problem, called maximum k-(set)-coverage problem. For this problem we present an algorithm that improves upon former results for the same model for small and moderate values of k.