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.

我们研究在线预算最大覆盖率问题。加权地面集U 的子集一个一个地到达，其中每个集都有一个成本。在线算法必须在其成本最多为给定预算的约束下选择一组集合。当一个集合到达时，算法必须决定是接受还是不可撤销地拒绝到达的集合，并且它也可以不可撤销地丢弃先前接受的集合。目标是最大化所选集合中集合所涵盖的元素的总权重。我们给出了一个确定性的\(\frac{4}{1-r}\) -竞争算法，其中r是一个集合的成本与总预算之间的最大比率，并表明任何确定性在线算法的竞争比率是\(\Omega (\frac{1}{1-r})\)。我们进一步给出了一个随机的O (1)-竞争算法。我们还给出了一个确定性的\(O(\Delta )\) -竞争算法，其中\(\Delta \)是一个集合的最大权重和它的一个修改版本，竞争比率为\(O(\min \{ \Delta ,\sqrt{w(U)}\})\)对于元素的总权重w ( U ) 预先已知的情况。\(\Omega (\min \{\Delta ,\sqrt{w(U)}\})\) 的匹配下界给出。最后，我们的结果，包括下限，也适用于Removable Online Knapsack。