Online Budgeted Maximum Coverage

We study the Online Budgeted Maximum Coverage problem. Subsets of a weighted ground set U arrive one by one, where each set has a cost. The online algorithm has to select a collection of sets, under the constraint that their cost is at most a given budget. Upon arrival of a set the algorithm must decide whether to accept or to irrevocably reject the arriving set, and it may also irrevocably drop previously accepted sets. The goal is to maximize the total weight of the elements covered by the sets in the chosen collection. We give a deterministic $$\frac{4}{1-r}$$ -competitive algorithm where r is the maximum ratio between the cost of a set and the total budget, and show that the competitive ratio of any deterministic online algorithm is $$\Omega (\frac{1}{1-r})$$ . We further give a randomized O(1)-competitive algorithm. We also give a deterministic $$O(\Delta )$$ -competitive algorithm, where $$\Delta $$ is the maximum weight of a set and a modified version of it with competitive ratio of $$O(\min \{\Delta ,\sqrt{w(U)}\})$$ for the case that the total weight of the elements, w(U), is known in advance. A matching lower bound of $$\Omega (\min \{\Delta ,\sqrt{w(U)}\})$$ is given. Finally, our results, including the lower bounds, apply also to Removable Online Knapsack.