## Abstract

In the classical Steiner tree problem, one is given an undirected, connected graph G = (V, E) with non-negative edge costs and a set of terminals T ⊆ V . The objective is to find a minimum-cost edge set E ⊆ E that spans the terminals. The problem is APX-hard; the best known approximation algorithm has a ratio of ρ = ln(4) + ε < 1.39. In this paper, we study a natural generalization, the multi-level Steiner tree (MLST) problem: given a nested sequence of terminals T_{1} ⊂ · · · ⊂ T_{k} ⊆ V , compute nested edge sets E_{1} ⊆ · · · ⊆ E_{k} ⊆ E that span the corresponding terminal sets with minimum total cost. The MLST problem and variants thereof have been studied under names such as Quality-of-Service Multicast tree, Grade-of-Service Steiner tree, and Multi-Tier tree. Several approximation results are known. We first present two natural heuristics with approximation factor O(k). Based on these, we introduce a composite algorithm that requires 2^{k} Steiner tree computations. We determine its approximation ratio by solving a linear program. We then present a method that guarantees the same approximation ratio and needs at most 2k Steiner tree computations. We compare five algorithms experimentally on several classes of graphs using four types of graph generators. We also implemented an integer linear program for MLST to provide ground truth.

Our combined algorithm outperforms the others both in theory and in practice when the number of levels is small (k ≤ 22), which works well for applications such as designing multi-level infrastructure or network visualization.

Original language | English (US) |
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Title of host publication | 17th Symposium on Experimental Algorithms, SEA 2018 |

Editors | Gianlorenzo D'Angelo |

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

Pages | 15:1-15:14 |

ISBN (Electronic) | 9783959770705 |

DOIs | |

State | Published - Jun 1 2018 |

Event | 17th Symposium on Experimental Algorithms, SEA 2018 - L'Aquila, Italy Duration: Jun 27 2018 → Jun 29 2018 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 103 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 17th Symposium on Experimental Algorithms, SEA 2018 |
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Country | Italy |

City | L'Aquila |

Period | 6/27/18 → 6/29/18 |

## Keywords

- Approximation algorithm
- Multi-level graph representation
- Steiner tree

## ASJC Scopus subject areas

- Software