## Abstract

A spanning tree T of graph G is a ρ-approximate universal Steiner tree (UST) for root vertex r if, for any subset of vertices S containing r, the cost of the minimal subgraph of T connecting S is within a ρ factor of the minimum cost tree connecting S in G. Busch et al. (FOCS 2012) showed that every graph admits 2O(√log n)-approximate USTs by showing that USTs are equivalent to strong sparse partition hierarchies (up to poly-logs). Further, they posed poly-logarithmic USTs and strong sparse partition hierarchies as open questions.We settle these open questions by giving polynomial-time algorithms for computing both O(log 7 n)-approximate USTs and poly-logarithmic strong sparse partition hierarchies. We reduce the existence of these objects to the previously studied cluster aggregation problem and a class of well-separated point sets which we call dangling nets. For graphs with constant doubling dimension or constant pathwidth we obtain improved bounds by deriving O(log n)-approximate USTs and O(1) strong sparse partition hierarchies. Our doubling dimension result is tight up to second order terms.

Original language | English |
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Title of host publication | Proceedings - 2023 IEEE 64th Annual Symposium on Foundations of Computer Science, FOCS 2023 |

Publisher | IEEE Computer Society |

Pages | 60-76 |

Number of pages | 17 |

ISBN (Electronic) | 9798350318944 |

DOIs | |

State | Published - 2023 |

Event | 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023 - Santa Cruz, United States Duration: 6 Nov 2023 → 9 Nov 2023 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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ISSN (Print) | 0272-5428 |

### Conference

Conference | 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023 |
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Country/Territory | United States |

City | Santa Cruz |

Period | 6/11/23 → 9/11/23 |

### Bibliographical note

Publisher Copyright:© 2023 IEEE.

### Funding

Busch supported by National Science Foundation grant CNS-2131538. Filtser supported by the Israel Science Foundation (grant No. 1042/22). Hathcock supported by the National Science Foundation Graduate Research Fellowship under grant No. DGE-2140739. Hershkowitz funded by the SNSF, Swiss National Science Foundation grant 200021 184622. Rajaraman supported by National Science Foundation grant CCF-1909363.

Funders | Funder number |
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National Science Foundation | CNS-2131538 |

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung | 200021 184622, CCF-1909363 |

Israel Science Foundation | DGE-2140739, 1042/22 |

## Keywords

- Steiner trees
- approximation algorithms
- metric embeddings
- universal algorithms