## Abstract

We present a unified (randomized) polynomial-time approximation scheme (PTAS) for the prize collecting traveling salesman problem (PCTSP) and the prize collecting Steiner tree problem (PCSTP) in doubling metrics. Given a metric space and a penalty function on a subset of points known as terminals, a solution is a subgraph on points in the metric space, whose cost is the weight of its edges plus the penalty due to terminals not covered by the subgraph. Under our unified framework, the solution subgraph needs to be Eulerian for PCTSP, while it needs to be a tree for PCSTP. Before our work, even a QPTAS for the problems in doubling metrics is not known. Our unified PTAS is based on the previous dynamic programming frameworks proposed in [Talwar STOC 2004] and [Bartal, Gottlieb, Krauthgamer STOC 2012]. However, since it is unknown which part of the optimal cost is due to edge lengths and which part is due to penalties of uncovered terminals, we need to develop new techniques to apply previous divide-and-conquer strategies and sparse instance decompositions.

Original language | English |
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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 |

ISBN (Print) | 9783959770811 |

DOIs | |

Publication status | Published - 1 Aug 2018 |

MoE publication type | A4 Article in a conference publication |

Event | European Symposia on Algorithms - Helsinki, Finland Duration: 20 Aug 2018 → 22 Aug 2018 Conference number: 26 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 112 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | European Symposia on Algorithms |
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Abbreviated title | ESA |

Country | Finland |

City | Helsinki |

Period | 20/08/2018 → 22/08/2018 |

## Keywords

- Doubling dimension
- Polynomial time approximation scheme
- Prize collecting
- Steiner tree problem
- Traveling salesman problem