## Abstract

A partition P of a weighted graph G is (σ, τ, ∆)-sparse if every cluster has diameter at most ∆, and every ball of radius ∆/σ intersects at most τ clusters. Similarly, P is (σ, τ, ∆)-scattering if instead for balls we require that every shortest path of length at most ∆/σ intersects at most τ clusters. Given a graph G that admits a (σ, τ, ∆)-sparse partition for all ∆ > 0, Jia et al. [STOC05] constructed a solution for the Universal Steiner Tree problem (and also Universal TSP) with stretch O(τσ^{2} log_{τ} n). Given a graph G that admits a (σ, τ, ∆)-scattering partition for all ∆ > 0, we construct a solution for the Steiner Point Removal problem with stretch O(τ^{3}σ^{3}). We then construct sparse and scattering partitions for various different graph families, receiving many new results for the Universal Steiner Tree and Steiner Point Removal problems.

Original language | English |
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Title of host publication | 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020 |

Editors | Artur Czumaj, Anuj Dawar, Emanuela Merelli |

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

ISBN (Electronic) | 9783959771382 |

DOIs | |

State | Published - 1 Jun 2020 |

Externally published | Yes |

Event | 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020 - Virtual, Online, Germany Duration: 8 Jul 2020 → 11 Jul 2020 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 168 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020 |
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Country/Territory | Germany |

City | Virtual, Online |

Period | 8/07/20 → 11/07/20 |

### Bibliographical note

Publisher Copyright:© Arnold Filtser; licensed under Creative Commons License CC-BY 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020).

## Keywords

- Scattering partitions
- Sparse covers
- Sparse partitions
- Steiner point removal
- Universal Steiner tree
- Universal TSP