Given a hypergraph $$\mathcal {H}=(X,{\mathcal {S}})$$ H = ( X , S ) , a planar support for $$\mathcal {H}$$ H is a planar graph G with vertex set X , such that for each hyperedge $$S\in \mathcal {S}$$ S ∈ S , the subgraph of G induced by the vertices in S is connected. Planar supports for hypergraphs have found several algorithmic applications, including several packing and covering problems, hypergraph coloring, and in hypergraph visualization. The main result proved in this paper is the following: given two families of regions R and B in the plane, each of which consists of connected, non-piercing regions, the intersection hypergraph $${\mathcal {H}}_R(B)=(B,\{B_{r}\}_{r\in R})$$ H R ( B ) = ( B , { B r } r ∈ R ) , where $$B_r=\{b\in B:b\cap r\ne \emptyset \}$$ B r = { b ∈ B : b ∩ r ≠ ∅ } has a planar support. Further, such a planar support can be computed in time polynomial in | R |, | B |, and the number of vertices in the arrangement of the regions in $$R\cup B$$ R ∪ B . Special cases of this result include the setting where either the family R , or the family B is a set of points. Our result unifies and generalizes several previous results on planar supports, PTAS’s for packing and covering problems on non-piercing regions in the plane and coloring of intersection hypergraph of non-piercing regions.

中文翻译：

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为非穿刺区域构建平面支撑

给定超图 $$\mathcal {H}=(X,{\mathcal {S}})$$ H = ( X , S ) ，$$\mathcal {H}$$ H 的平面支持是平面图G 具有顶点集 X ，使得对于每个超边 $$S\in \mathcal {S}$$ S ∈ S ，由 S 中的顶点诱导的 G 的子图是连通的。对超图的平面支持已经发现了几种算法应用，包括几个打包和覆盖问题、超图着色和超图可视化。本文证明的主要结果如下：给定平面上的两个区域 R 和 B 族，每个族都由相连的非穿透区域组成，交集超图 $${\mathcal {H}}_R(B )=(B,\{B_{r}\}_{r\in R})$$ HR ( B ) = ( B , { B r } r ∈ R ) ，其中 $$B_r=\{b\in B:b\cap r\ne \emptyset \}$$ B r = { b ∈ B : b ∩ r ≠ ∅ } 具有平面支撑。更多，这样的平面支撑可以在时间多项式中计算 | R |, | B |，以及 $$R\cup B$$ R ∪ B 中区域排列的顶点数。此结果的特殊情况包括家庭 R 或家庭 B 是一组点的设置。我们的结果统一和概括了先前关于平面支撑、PTAS 的几个结果，用于包装和覆盖平面中非穿孔区域的问题以及非穿孔区域的交叉超图的着色。