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