We consider the Max Unique Coverage problem, including applications to the data stream model. The input is a universe of $n$ elements, a collection of $m$ subsets of this universe, and a cardinality constraint, $k$. The goal is to select a subcollection of at most $k$ sets that maximizes unique coverage, i.e, the number of elements contained in exactly one of the selected sets. The Max Unique Coverage problem has applications in wireless networks, radio broadcast, and envy-free pricing. Our first main result is a fixed-parameter tractable approximation scheme (FPT-AS) for Max Unique Coverage, parameterized by $k$ and the maximum element frequency, $r$, which can be implemented on a data stream. Our FPT-AS finds a $(1-\epsilon)$-approximation while maintaining a kernel of size $\tilde{O}(k r/\epsilon)$, which can be combined with subsampling to use $\tilde{O}(k^2 r / \epsilon^3)$ space overall. This significantly improves on the previous-best FPT-AS with the same approximation, but a kernel of size $\tilde{O}(k^2 r / \epsilon^2)$. In order to achieve our result, we show upper bounds on the ratio of a collection's coverage to the unique coverage of a maximizing subcollection; this is by constructing explicit algorithms that find a subcollection with unique coverage at least a logarithmic ratio of the collection's coverage. We complement our algorithms with our second main result, showing that $\Omega(m / k^2)$ space is necessary to achieve a $(1.5 + o(1))/(\ln k - 1)$-approximation in the data stream. This dramatically improves the previous-best lower bound showing that $\Omega(m / k^2)$ is necessary to achieve better than a $e^{-1+1/k}$-approximation.

翻译：我们研究最大唯一覆盖问题，包括其在数据流模型中的应用。输入包含一个大小为$n$的元素全集、该全集上$m$个子集的集合，以及基数约束$k$。目标是选择最多$k$个子集，使得唯一覆盖最大化，即恰好被一个选中子集包含的元素数量。最大唯一覆盖问题在无线网络、无线电广播和无嫉妒定价中具有应用。我们的第一个主要结果是针对最大唯一覆盖的固定参数可处理近似方案（FPT-AS），参数为$k$与最大元素频率$r$，该方案可在数据流上实现。我们的FPT-AS在保持$\tilde{O}(k r/\epsilon)$大小核的同时，找到$(1-\epsilon)$近似解，结合子采样技术后总体空间使用为$\tilde{O}(k^2 r / \epsilon^3)$。这显著改进了此前具有相同近似比但核大小为$\tilde{O}(k^2 r / \epsilon^2)$的最佳FPT-AS。为实现该结果，我们通过构造显式算法证明了集合覆盖值与最大化子集唯一覆盖值的比值上界，该算法可找到唯一覆盖至少达集合覆盖值对数比例的子集。我们通过第二个主要结果补充了算法分析，证明在数据流中实现$(1.5 + o(1))/(\ln k - 1)$近似必须使用$\Omega(m / k^2)$空间。这显著改进了先前最佳下界结果——原结果仅证明要实现优于$e^{-1+1/k}$的近似必须使用$\Omega(m / k^2)$空间。