We revisit the classic Maximum $k$-Coverage problem: Determine the largest number $t$ of elements that can be covered by choosing $k$ sets from a given family $\mathcal{F} = \{S_1,\dots, S_n\}$ of a size-$u$ universe. A notable special case is Partial $k$-Dominating Set, where one chooses $k$ vertices in a graph to maximize the number of dominated vertices. Extensive research has established strong hardness results for various aspects of Maximum $k$-Coverage, such as tight inapproximability results, $W[2]$-hardness, and a conditionally tight worst-case running time of $n^{k\pm o(1)}$. In this paper we ask: (1) Can this time bound be improved for small $t$, at least for Partial $k$-Dominating Set, ideally to time~$t^{k\pm O(1)}$? (2) More ambitiously, can we even determine the best-possible running time of Maximum $k$-Coverage with respect to the perhaps most natural parameters: the universe size $u$, the maximum set size $s$, and the maximum frequency $f$? We successfully resolve both questions. (1) We give an algorithm that solves Partial $k$-Dominating Set in time $O(nt + t^{\frac{2ω}{3} k+O(1)})$ if $ω\ge 2.25$ and time $O(nt+ t^{\frac{3}{2} k+O(1)})$ if $ω\le 2.25$, where $ω\le 2.372$ is the matrix multiplication exponent. From this we derive a time bound that is conditionally optimal, regardless of $ω$, based on the well-established $k$-clique and 3-uniform hyperclique hypotheses from fine-grained complexity. We also obtain matching upper and lower bounds for sparse graphs. To address (2) we design an algorithm for Maximum $k$-Coverage running in time $$ \min \left\{ (f\cdot \min\{\sqrt[3]{u}, \sqrt{s}\})^k + \min\{n,f\cdot \min\{\sqrt{u}, s\}\}^{kω/3}, n^k\right\} \cdot g(k)n^{\pm O(1)}, $$ and, surprisingly, further show that this complicated time bound is also conditionally optimal.

翻译：我们重新审视经典的最大$k$覆盖问题：给定一个定义在大小为$u$的全集上的集合族$\mathcal{F} = \{S_1,\dots, S_n\}$，选取$k$个集合，求所能覆盖元素的最大数量$t$。一个值得注意的特例是部分$k$支配集问题，即在图中选取$k$个顶点，以最大化被支配顶点的数量。已有大量研究针对最大$k$覆盖问题的各个方面建立了强硬度结果，例如紧不可近似性结果、$W[2]$硬度，以及条件紧的最坏情况运行时间$n^{k\pm o(1)}$。本文我们提出两个问题：(1) 对于较小的$t$值，至少对于部分$k$支配集问题，能否改进这一时间界，理想情况下达到$t^{k\pm O(1)}$？(2) 更宏大地，我们能否针对最大$k$覆盖问题，基于最自然的参数——全集大小$u$、最大集合大小$s$和最大频率$f$——确定其最佳可能运行时间？我们成功解决了这两个问题。(1) 我们给出一个算法，若矩阵乘法指数$ω\ge 2.25$，该算法以$O(nt + t^{\frac{2ω}{3} k+O(1)})$的时间求解部分$k$支配集；若$ω\le 2.25$，则以$O(nt+ t^{\frac{3}{2} k+O(1)})$的时间求解，其中$ω\le 2.372$。由此我们导出一个条件最优的时间界（与$ω$无关），其依据是细粒度复杂性理论中已确立的$k$团假设和3一致超团假设。我们还得到了稀疏图上的匹配上下界。针对问题(2)，我们设计了一个求解最大$k$覆盖的算法，其运行时间为 $$ \min \left\{ (f\cdot \min\{\sqrt[3]{u}, \sqrt{s}\})^k + \min\{n,f\cdot \min\{\sqrt{u}, s\}\}^{kω/3}, n^k\right\} \cdot g(k)n^{\pm O(1)}, $$ 并且令人惊讶地，我们进一步证明了这个复杂的时间界同样是条件最优的。