## Abstract

Let F be a family of pseudo-disks in the plane, and P be a finite subset of F. Consider the hyper-graph H(P,F) whose vertices are the pseudo-disks in P and the edges are all subsets of P of the form {D∈P|D∩S≠∅}, where S is a pseudo-disk in F. We give an upper bound of O(nk^{3}) for the number of edges in H(P,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.

Original language | English |
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Article number | 101687 |

Journal | Computational Geometry: Theory and Applications |

Volume | 92 |

DOIs | |

State | Published - Jan 2021 |

### Bibliographical note

Publisher Copyright:© 2020 Elsevier B.V.

### Funding

Work on this paper by Boris Aronov has been supported by NSA MSP Grant H98230-10-1-0210, by NSF Grants CCF-08-30691, CCF-11-17336, CCF-12-18791, and CCF-15-40656, and by U.S.-Israel Binational Science Foundation grant 2014/170.Work on this paper by Anirudh Donakonda has been partially supported by NSF Grant CCF-11-17336.Work on this paper by Esther Ezra has been supported by ISF Grant 824/17, NSF under grants CAREER CCF-15-53354, CCF-11-17336, and CCF-12-16689.Work on this paper by Rom Pinchasi has been supported by ISF grant No. 409/16.

Funders | Funder number |
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NSA MSP | H98230-10-1-0210 |

U.S.-Israel Binational Science Foundation | 2014/170 |

National Science Foundation | CCF-08-30691, CCF-11-17336, CCF-12-18791, CCF-15-40656 |

Iowa Science Foundation | CCF-12-16689, 824/17, CCF-15-53354, 409/16 |

## Keywords

- Approximation algorithms
- Hypergraphs of finite VC dimension
- Minimum-weight dominating set
- Planar graphs
- Pseudo-disks