## Abstract

This paper examines a number of variants of the sparse k- spanner problem, and presents hardness results concerning their approximability. Previously, it was known that most k-spanner problems are weakly inapproximable, namely, are NP-hard to approximate with ratio O(log n), for every k ≥ 2, and that the unit-length k-spanner problem for constant stretch requirement k ≥ 5 is strongly inapproximable, namely, is NP-hard to approximate with ratio O(2^{log}^{Ɛn}) [19]. The results of this paper significantly expand the ranges of hardness for k-spanner problems. In general, strong hardness is shown for a number of k-spanner problems, for certain ranges of the stretch requirement k depending on the particular variant at hand. The problems studied differ by the types of edge weights and lengths used, and include also directed, augmentation and client-server variants of the problem. The paper also considers k-spanner problems in which the stretch requirement k is relaxed (e.g., k =Ω(log n)). For these cases, no inapproximability results were known at all (even for a constant approximation ratio) for any spanner problem. Moreover, some versions of the k-spanner problem are known to enjoy the ratio degradation property, namely, their complexity decreases exponentially with the inverse of the stretch requirement. So far, no hardness result existed precluding any k-spanner problem from enjoying this property. This paper establishes strong inapproximability results for the case of relaxed stretch requirement (up to k = o(n^{δ}), for any 0 < δ < 1), for a large variety of k-spanner problems. It is also shown that these problems do not enjoy the ratio degradation property.

Original language | English |
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Title of host publication | STACS 2000 - 17th Annual Symposium on Theoretical Aspects of Computer Science, STACS 2000, Proceedings |

Editors | Horst Reichel, Sophie Tison |

Publisher | Springer Verlag |

Pages | 370-381 |

Number of pages | 12 |

ISBN (Print) | 9783540671411 |

DOIs | |

State | Published - 2000 |

Externally published | Yes |

Event | 17th Annual Symposium on Theoretical Aspects of Computer Science, STACS 2000 - Lille, France Duration: 17 Feb 2000 → 19 Feb 2000 |

### 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 | 1770 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 17th Annual Symposium on Theoretical Aspects of Computer Science, STACS 2000 |
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Country/Territory | France |

City | Lille |

Period | 17/02/00 → 19/02/00 |

### Bibliographical note

Publisher Copyright:© Springer-Verlag Berlin Heidelberg 2000.