We study the discrete bin covering problem where a multiset of items from a fixed set $S \subseteq (0,1]$ must be split into disjoint subsets while maximizing the number of subsets whose contents sum to at least $1$. We study the online discrete variant, where $S$ is finite, and items arrive sequentially. In the purely online setting, we show that the competitive ratios of best deterministic (and randomized) algorithms converge to $\frac{1}{2}$ for large $S$, similar to the continuous setting. Therefore, we consider the problem under the prediction setting, where algorithms may access a vector of frequencies predicting the frequency of items of each size in the instance. In this setting, we introduce a family of online algorithms that perform near-optimally when the predictions are correct. Further, we introduce a second family of more robust algorithms that presents a tradeoff between the performance guarantees when the predictions are perfect and when predictions are adversarial. Finally, we consider a stochastic setting where items are drawn independently from any fixed but unknown distribution of $S$. Using results from the PAC-learnability of probabilities in discrete distributions, we also introduce a purely online algorithm whose average-case performance is near-optimal with high probability for all finite sets $S$ and all distributions of $S$.

翻译：我们研究离散装箱覆盖问题，其中来自固定集合$S \subseteq (0,1]$的多重集物品必须被划分为不相交的子集，同时最大化子集内容之和至少为$1$的子集数量。我们研究在线离散变体，其中$S$是有限的，且物品按顺序到达。在纯在线设置中，我们证明最佳确定性（及随机化）算法的竞争比对于大的$S$收敛于$\frac{1}{2}$，与连续设置类似。因此，我们在预测设置下考虑该问题，其中算法可以访问频率向量，预测实例中每种尺寸物品的频率。在此设置中，我们引入一系列在线算法，当预测正确时其性能接近最优。此外，我们引入第二个更鲁棒的算法系列，在预测完美和预测对抗的情况下，呈现出性能保证之间的权衡。最后，我们考虑一个随机设置，其中物品独立地从$S$的任意固定但未知分布中抽取。利用离散分布中概率的PAC可学习性结果，我们还引入一个纯在线算法，对于所有有限集合$S$和所有$S$的分布，其平均情况性能以高概率接近最优。