## Abstract

Directed Steiner Tree (DST) is a central problem in combinatorial optimization and theoretical computer science: Given a directed graph G= (V, E) with edge costs c∈R≥0E, a root r∈ V and k terminals K⊆ V, we need to output the minimum-cost arborescence in G that contains an r→ t path for every t∈ K. Recently, Grandoni, Laekhanukit and Li, and independently Ghuge and Nagarajan, gave quasi-polynomial time O(log ^{2}k/ log log k) -approximation Algorithms for the problem, which are tight under popular complexity assumptions. In this paper, we consider the more general Degree-Bounded Directed Steiner Tree (DB-DST) problem, where we are additionally given a degree bound d_{v} on each vertex v∈ V, and we require that every vertex v in the output tree has at most d_{v} children. We give a quasi-polynomial time (O(log nlog k) , O(log ^{2}n)) -bicriteria approximation: The Algorithm produces a solution with cost at most O(log nlog k) times the cost of the optimum solution that violates the degree constraints by at most a factor of O(log ^{2}n). This is the first non-trivial result for the problem. While our cost-guarantee is nearly optimal, the degree violation factor of O(log ^{2}n) is an O(log n) -factor away from the approximation lower bound of Ω (log n) from the set-cover hardness. The hardness result holds even on the special case of the Degree-Bounded Group Steiner Tree problem on trees (DB-GST-T). With the hope of closing the gap, we study the question of whether the degree violation factor can be made tight for this special case. We answer the question in the affirmative by giving an (O(log nlog k) , O(log n)) -bicriteria approximation Algorithm for DB-GST-T.

Original language | English |
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Article number | 5 |

Pages (from-to) | 1252-1278 |

Number of pages | 27 |

Journal | Algorithmica |

Volume | 84 |

Issue number | 5 |

Early online date | 24 Jan 2022 |

DOIs | |

State | Published - May 2022 |

Externally published | Yes |

### Bibliographical note

Funding Information:X. Guo, S. Li and J. Xian are partially supported by NSF grants CCF-1566356, CCF- 1717134, CCF-1844890. B. Laekhanukit is partially supported by Science and Technology Innovation 2030 -“New Generation of Artificial Intelligence” Major Project No.(2018AAA0100903), NSFC grant 61932002, Program for Innovative Research Team of Shanghai University of Finance and Economics (IRTSHUFE) and the Fundamental Research Funds for the Central Universities and by the 1000-talent award by the Chinese Government. Daniel Vaz has been supported by the Alexander von Humboldt Foundation with funds from the German Federal Ministry of Education and Research (BMBF). G. Kortsarz is partially supported by NSF grant 1910565.

Publisher Copyright:

© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

## Keywords

- Approximation
- Degrees
- Group
- Steiner