## Abstract

Motivated by some open problems posed in [13], we study three problems that seek a low degree subtree T of a graph G =(V, E). In the Min-Degree Group Steiner Tree problem we are given a collection of node subsets (groups), and T should contain a node from every group. In the Min-Degree Steiner k-Tree problem we are given a set R of terminals and an integer k, and T should contain k terminals. In both problems the goal is to minimize the maximum degree of T . In the more general Degrees Bounded Min-Cost Group Steiner Tree problem, we are also given edge costs and individual degree bounds (Formula Presented). The output tree T should obey the degree constraints deg_{T} (v) ≤ b_{v} for all (Formula Presented), and among all such trees we seek one of minimum cost. When the input is a tree, an O(log^{2} n) approximation for the cost is given in [10]. Our first result generalizes [10] – we give a bicriteria (O(log^{2} n), O(log^{2} n))-approximation algorithm for Degrees Bounded Min-Cost Group Steiner Tree problem on tree inputs. This matches the cost ratio of [10] but also approximates the degrees within O(log^{2} n). Our second result shows that if Min-Degree Group Steiner Tree admits ratio ρ then Min-Degree Steiner k-Tree admits ratio ρ · O(log k). Combined with [12], this implies an O(log^{3} n)-approximation for Min-Degree Steiner k-Tree on general graphs, in quasi-polynomial time. Our third result is a polynomial time O(log^{3} n)-approximation algorithm for Min-Degree Group Steiner Tree on bounded treewidth graphs.

Original language | English |
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Title of host publication | Combinatorial Algorithms - 31st International Workshop, IWOCA 2020, Proceedings |

Editors | Leszek Gasieniec, Leszek Gasieniec, Ralf Klasing, Tomasz Radzik |

Publisher | Springer |

Pages | 343-354 |

Number of pages | 12 |

ISBN (Print) | 9783030489656 |

DOIs | |

State | Published - 2020 |

Event | 31st International Workshop on Combinatorial Algorithms, IWOCA 2020 - Bordeaux, France Duration: 8 Jun 2020 → 10 Jun 2020 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 12126 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 31st International Workshop on Combinatorial Algorithms, IWOCA 2020 |
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Country/Territory | France |

City | Bordeaux |

Period | 8/06/20 → 10/06/20 |

### Bibliographical note

Publisher Copyright:© Springer Nature Switzerland AG 2020.