## 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, they 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, it is NP-hard to approximate with ratio O(2^{log1-εn}))[27]. 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 they include directed, augmentation and client-server variants. 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 (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^{1-σ}), for any 0 < σ < 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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Pages (from-to) | 691-729 |

Number of pages | 39 |

Journal | Theory of Computing Systems |

Volume | 41 |

Issue number | 4 |

DOIs | |

State | Published - Dec 2007 |

Externally published | Yes |

### Bibliographical note

Funding Information:∗ This work by Michael Elkin was done in the Weizmann Institute of Science, Rehovot, Israel. David Peleg was supported in part by grants from the Israel Science Foundation and the Israel Ministry of Science and Art.

### Funding

∗ This work by Michael Elkin was done in the Weizmann Institute of Science, Rehovot, Israel. David Peleg was supported in part by grants from the Israel Science Foundation and the Israel Ministry of Science and Art.

Funders | Funder number |
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Israel Ministry of Science and Art | |

Israel Science Foundation |