### Abstract

Given a hypergraph H = (X, S), a planar support for H is a planar graph G with vertex set X, such that for each hyperedge S ∈ S, the sub-graph 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 H_{R}(B) = (B,{B_{r}}_{r∈R}), where B_{r} = {b ∈ B: b ∩ r ≠ θ} 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 ∪ 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, PTASs for packing and covering problems on non-piercing regions in the plane and coloring of intersection hypergraph of non-piercing regions.

Original language | English (US) |
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Title of host publication | 26th European Symposium on Algorithms, ESA 2018 |

Editors | Hannah Bast, Grzegorz Herman, Yossi Azar |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Volume | 112 |

ISBN (Print) | 9783959770811 |

DOIs | |

State | Published - Aug 1 2018 |

Event | 26th European Symposium on Algorithms, ESA 2018 - Helsinki, Finland Duration: Aug 20 2018 → Aug 22 2018 |

### Other

Other | 26th European Symposium on Algorithms, ESA 2018 |
---|---|

Country | Finland |

City | Helsinki |

Period | 8/20/18 → 8/22/18 |

### Fingerprint

### Keywords

- Geometric optimization
- Non-piercing regions
- Packing and covering

### ASJC Scopus subject areas

- Software

### Cite this

*26th European Symposium on Algorithms, ESA 2018*(Vol. 112). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ESA.2018.69

**Planar support for non-piercing regions and applications.** / Raman, Rajiv; Ray, Saurabh.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*26th European Symposium on Algorithms, ESA 2018.*vol. 112, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 26th European Symposium on Algorithms, ESA 2018, Helsinki, Finland, 8/20/18. https://doi.org/10.4230/LIPIcs.ESA.2018.69

}

TY - GEN

T1 - Planar support for non-piercing regions and applications

AU - Raman, Rajiv

AU - Ray, Saurabh

PY - 2018/8/1

Y1 - 2018/8/1

N2 - Given a hypergraph H = (X, S), a planar support for H is a planar graph G with vertex set X, such that for each hyperedge S ∈ S, the sub-graph 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 HR(B) = (B,{Br}r∈R), where Br = {b ∈ B: b ∩ r ≠ θ} 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 ∪ 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, PTASs for packing and covering problems on non-piercing regions in the plane and coloring of intersection hypergraph of non-piercing regions.

AB - Given a hypergraph H = (X, S), a planar support for H is a planar graph G with vertex set X, such that for each hyperedge S ∈ S, the sub-graph 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 HR(B) = (B,{Br}r∈R), where Br = {b ∈ B: b ∩ r ≠ θ} 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 ∪ 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, PTASs for packing and covering problems on non-piercing regions in the plane and coloring of intersection hypergraph of non-piercing regions.

KW - Geometric optimization

KW - Non-piercing regions

KW - Packing and covering

UR - http://www.scopus.com/inward/record.url?scp=85052498853&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85052498853&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ESA.2018.69

DO - 10.4230/LIPIcs.ESA.2018.69

M3 - Conference contribution

SN - 9783959770811

VL - 112

BT - 26th European Symposium on Algorithms, ESA 2018

A2 - Bast, Hannah

A2 - Herman, Grzegorz

A2 - Azar, Yossi

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -