## Abstract

We provide the first hardness result of a polylogarithmic approximation ratio for a natural NP-hard optimization problem. We show that for every fixed ε > 0, the GROUP-STEINER-TREE problem admits no efficient log^{2-ε} k approximation, where k denotes the number of groups (or, alternatively, the input size), unless NP has quasi-polynomial Las-Vegas algorithms. This hardness result holds even for input graphs which are Hierarchically Well-Separated Trees, introduced by Bartal [FOCS, 1996]. For these trees (and also for general trees), our bound is nearly tight with the log-squared approximation currently known. Our results imply that for every fixed ε > 0, the DIRECTED-STEINER-TREE problem admits no log^{2-ε} n-approximation, where n is the number of vertices in the graph, under the same complexity assumption.

Original language | English |
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Pages (from-to) | 585-594 |

Number of pages | 10 |

Journal | Conference Proceedings of the Annual ACM Symposium on Theory of Computing |

DOIs | |

State | Published - 2003 |

Externally published | Yes |

Event | 35th Annual ACM Symposium on Theory of Computing - San Diego, CA, United States Duration: 9 Jun 2003 → 11 Jun 2003 |

## Keywords

- Approximation algorithms
- Hardness of approximation
- Integrality ratio
- Polylogarithmic approximation
- Steiner tree