Let $\mathcal{F}$ be a family of pseudo-disks in the plane, and $\mathcal{P}$ be a finite subset of $\mathcal{F}$. Consider the hyper-graph $H(\mathcal{P},\mathcal{F})$ whose vertices are the pseudo-disks in $\mathcal{P}$ and the edges are all subsets of $\mathcal{P}$ of the form $\{D\in \mathcal{P}|D\cap S\ne \varnothing \}$, where S is a pseudo-disk in $\mathcal{F}$. We give an upper bound of $O(n{k}^3)$ for the number of edges in $H(\mathcal{P},\mathcal{F})$ of cardinality at most k. This generalizes a result of Buzaglo et al. [4].

As an application of our bound, we obtain an algorithm that computes a constant-factor approximation to the minimum-weight dominating set in a collection of pseudo-disks in the plane, in expected polynomial time.

中文翻译：

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在伪磁盘超图上

让 $\mathcal{F}$ 成为平面中的伪磁盘族，并且 $\mathcal{P}$ 是...的有限子集 $\mathcal{F}$。考虑超图$H（\mathcal{P}，\mathcal{F}）$ 其顶点是其中的伪磁盘 $\mathcal{P}$ 边缘都是的子集 $\mathcal{P}$ 的形式 $\{d\in \mathcal{P}|d\cap \mathrm{小号}\ne \varnothing \}$，其中S是位于中的伪磁盘$\mathcal{F}$。我们给的上限$Ø（ñ{ķ}^3）$ 对于中的边数 $H（\mathcal{P}，\mathcal{F}）$基数最大为k。这概括了Buzaglo等人的结果。[4]。

作为边界的一种应用，我们获得了一种算法，该算法可以在期望的多项式时间内计算出平面中伪磁盘集合中最小权重控制集的常数因子近似值。