The maximum coverage problem is to select $k$ sets from a collection of sets such that the cardinality of the union of the selected sets is maximized. We consider $(1-1/e-\epsilon)$-approximation algorithms for this NP-hard problem in three standard data stream models. 1. {\em Dynamic Model.} The stream consists of a sequence of sets being inserted and deleted. Our multi-pass algorithm uses $\epsilon^{-2} k \cdot \text{polylog}(n,m)$ space. The best previous result (Assadi and Khanna, SODA 2018) used $(n +\epsilon^{-4} k) \text{polylog}(n,m)$ space. While both algorithms use $O(\epsilon^{-1} \log n)$ passes, our analysis shows that when $\epsilon$ is a constant, it is possible to reduce the number of passes by a $1/\log \log n$ factor without incurring additional space. 2. {\em Random Order Model.} In this model, there are no deletions and the sets forming the instance are uniformly randomly permuted to form the input stream. We show that a single pass and $k \text{polylog}(n,m)$ space suffices for arbitrary small constant $\epsilon$. The best previous result, by Warneke et al.~(ESA 2023), used $k^2 \text{polylog}(n,m)$ space. 3. {\em Insert-Only Model.} Lastly, our results, along with numerous previous results, use a sub-sampling technique introduced by McGregor and Vu (ICDT 2017) to sparsify the input instance. We explain how this technique and others used in the paper can be implemented such that the amortized update time of our algorithm is polylogarithmic. This also implies an improvement of the state-of-the-art insert only algorithms in terms of the update time: $\text{polylog}(m,n)$ update time suffices whereas the best previous result by Jaud et al.~(SEA 2023) required update time that was linear in $k$.

翻译：最大覆盖问题是从集合族中选择$k$个集合，使得所选集合并集的基数最大化。针对这一NP-hard问题，我们在三种标准数据流模型下研究$(1-1/e-\epsilon)$近似算法。1. **动态模型**：数据流由一系列集合的插入和删除操作组成。我们的多遍算法使用$\epsilon^{-2} k \cdot \text{polylog}(n,m)$空间，而先前最优结果（Assadi 和 Khanna, SODA 2018）需要$(n +\epsilon^{-4} k) \text{polylog}(n,m)$空间。虽然两种算法均使用$O(\epsilon^{-1} \log n)$遍，但我们的分析表明，当$\epsilon$为常数时，可在不增加空间开销的情况下将遍数减少$1/\log \log n$倍。2. **随机顺序模型**：该模型中无删除操作，构成实例的集合以均匀随机排列方式形成输入流。我们证明单遍扫描结合$k \text{polylog}(n,m)$空间即可满足任意小的常数$\epsilon$，而先前最优结果（Warneke 等, ESA 2023）需要$k^2 \text{polylog}(n,m)$空间。3. **仅插入模型**：最后，我们的结果与众多先前工作均采用McGregor 和 Vu (ICDT 2017) 提出的子采样技术对输入实例进行稀疏化。我们阐述了如何实现该技术及文中其他技术，使得算法的平摊更新时间为多对数级。这也意味着在更新时间复杂度上改进了当前最优的仅插入算法：仅需$\text{polylog}(m,n)$更新时间，而先前最优结果（Jaud 等, SEA 2023）所需更新时间与$k$呈线性关系。