Online maximum k-coverage
Résumé
We study an online model for the maximum $k$-vertex-coverage problem, in which, given a graph $G=(V,E)$ and an integer $k$, we seek a subset $A \subseteq V$ such that $\vert A \vert=k$ and the number of edges covered by $A$ is maximized. In our model, at each step $i$, a new vertex $v_i$ is released, 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 released is definitely deleted). We propose algorithms for several natural classes of graphs (mainly regular and bipartite), improving on an easy $\frac{1}{2}$-competitive ratio. We next settle a set version of the problem, called the 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 $k$.