## Abstract

A k-spanner of a connected graph G = (V, E) is a subgraph Gʹ consisting of all the vertices of V and a subset of the edges, with the additional property that the distance between any two vertices in Gʹ is larger than that distance in G by no more than a factor of k. This paper concerns the hardness of finding spanners with the number of edges close to the optimum. It is proved that for every fixed k approximating the spanner problem is at least as hard as approximating the set cover problem We also consider a weighted version of the spanner problem. We prove that in the case k = 2 the problem admits an O(log n)-ratio approximation, and in the case k ≥ 5, there is no 2^{log1-ϵ n}-ratio approximation, for any ϵ > 0, unless NP ⊆ DTIME(n^{polylog n}).

Original language | English |
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Title of host publication | Approximation Algorithms for Combinatorial Optimization - International Workshop, APPROX 1998, Proceedings |

Editors | José Rolim, Klaus Jansen |

Publisher | Springer Verlag |

Pages | 135-146 |

Number of pages | 12 |

ISBN (Print) | 3540647368, 9783540647362 |

DOIs | |

State | Published - 1998 |

Event | International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 1998 - Aalborg, Denmark Duration: 18 Jul 1998 → 19 Jul 1998 |

### 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 | 1444 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 1998 |
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Country/Territory | Denmark |

City | Aalborg |

Period | 18/07/98 → 19/07/98 |

### Bibliographical note

Publisher Copyright:© Springer-Verlag Berlin Heidelberg 1998.