## Abstract

We study two related network design problems with two cost functions. In the buy-at-bulk k-Steiner tree problem we are given a graph G(V, E) with a set of terminals T ⊆ V including a particular vertex s called the root, and an integer k ≤ |T|. There are two cost functions on the edges of G, a buy cost b : E → ℝ^{+} and a distance cost r : E → ℝ^{+}. The goal is to find a subtree H of G rooted at s with at least k terminals so that the cost ∑_{e∈H}b(e) + ∑_{t∈T-s} dist(t, s) is minimize, where dist(t,s) is the distance from t to s in H with respect to the r cost. We present an O(log ^{4} n)-approximation for the buy-at-bulk k-Steiner tree problem. The second and closely related one is bicriteria approximation algorithm for Shallow-light k-Steiner trees. In the shallow-light k-Steiner tree problem we are given a graph G with edge costs b(e) and distance costs r(e) over the edges, and an integer k. Our goal is to find a minimum cost (under b-cost) k-Steiner tree such that the diameter under r-cost is at most some given bound D. We develop an (O(log n), O(log^{3} n))-approximation algorithm for a relaxed version of Shallow-light k-Steiner tree where the solution has at least k/8 terminals. Using this we obtain an (O(log^{2} n), O(log^{4} n))-approximation for the shallow-light k-Steiner tree and an O(log^{4} n)-approximation for the buy-at-bulk k-Steiner tree problem.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 a |

Publisher | Springer Verlag |

Pages | 152-163 |

Number of pages | 12 |

ISBN (Print) | 3540380442, 9783540380443 |

DOIs | |

State | Published - 2006 |

Externally published | Yes |

Event | 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 and 10th International Workshop on Randomization and Computation, RANDOM 2006 - Barcelona, Spain Duration: 28 Aug 2006 → 30 Aug 2006 |

### 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 | 4110 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 and 10th International Workshop on Randomization and Computation, RANDOM 2006 |
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Country/Territory | Spain |

City | Barcelona |

Period | 28/08/06 → 30/08/06 |