We study the Online Budgeted Maximum Coverage (OBMC) 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 reject the arriving set, and it may also drop previously accepted sets (preemption). Rejecting or dropping a set is irrevocable. The goal is to maximize the total weight of the elements covered by the sets in the chosen collection. We present a deterministic 4/(1-r)-competitive algorithm for OBMC, where r is the maximum ratio between the cost of a set and the total budget. Building on that algorithm, we then present a randomized O(1)-competitive algorithm for OBMC. On the other hand, we show that the competitive ratio of any deterministic online algorithm is Omega(1/(sqrt{1-r})). We also give a deterministic O(Delta)-competitive algorithm, where Delta is the maximum weight of a set (given that the minimum element weight is 1), and if the total weight of all elements, w(U), is known in advance, we show that a slight modification of that algorithm is O(min{Delta,sqrt{w(U)}})-competitive. A matching lower bound of Omega(min{Delta,sqrt{w(U)}}) is also given. Previous to the present work, only the unit cost version of OBMC was studied under the online setting, giving a 4-competitive algorithm [Saha, Getoor, 2009]. Finally, our results, including the lower bounds, apply to Removable Online Knapsack which is the preemptive version of the Online Knapsack problem.
This is a joint work with Adi Rosén. |