## Abstract

Given a graph G = (V;E), a subgraph G’ = (V;H), H ⊆ E is a k-spanner (respectively, k-DSS) of G if for any pair of vertices u;w 2 V it satises dH(u;w) 2286; k 2286; dG(u;w) (resp., dH(u;w) 2286; k). The basic k-spanner (resp., k-DSS ) problem is to nd a k-spanner (resp., k-DSS) of a given graph G with the smallest possible number of edges. This paper considers approximation algorithms for these and some related problems for k > 2. Both problems are known to be 3A9; (2log1− ɛ n)-inapproximable [11,13]. The basic k-spanner problem over undirected graphs with k > 2 has been given a sublinear ratio approximation algorithm (with ratio roughly O(n ^{2/k+1}), but no such algorithms were known for other variants of the problem, including the directed and the clientserver variants, as well as for the k-DSS problem. We present the first approximation algorithms for these problems with sublinear approximation ratio. The second contribution of this paper is in characterizing some wide families of graphs on which the problems do admit a logarithmic and a polylogarithmic approximation ratios. These families are characterized as containing graphs that have optimal or \near-optimal" spanners with certain desirable properties, such as being a tree, having low arboricity or having low girth. All our results generalize to the directed and the client-server variants of the problems. As a simple corollary, we present an algorithm that given a graph G builds a subgraph with O (n) edges and stretch bounded by the tree-stretch of G, namely the minimum maximal stretch of a spanning tree for G. The analysis of our algorithms involves the novel notion of edge-dominating systems developed in the paper. The technique introduced in the paper enables us to reduce the studied algorithmic questions of approximability of the k-spanner and k-DSS problems to purely graph-theoretical questions concerning the existence of certain combinatorial objects in families of graphs.

Original language | English |
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Title of host publication | Integer Programming and Combinatorial Optimization - 8th International IPCO Conference, Proceedings |

Editors | Karen Aardal, Bert Gerards |

Publisher | Springer Verlag |

Pages | 90-104 |

Number of pages | 15 |

ISBN (Print) | 3540422250, 9783540422259 |

DOIs | |

State | Published - 2001 |

Externally published | Yes |

Event | 8th International Integer Programming and Combinatorial Optimization Conference, IPCO 2001 - Utrecht, Netherlands Duration: 13 Jun 2001 → 15 Jun 2001 |

### 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 | 2081 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 8th International Integer Programming and Combinatorial Optimization Conference, IPCO 2001 |
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Country/Territory | Netherlands |

City | Utrecht |

Period | 13/06/01 → 15/06/01 |

### Bibliographical note

Publisher Copyright:© Springer-Verlag Berlin Heidelberg 2001.

### Funding

∗Corresponding author. E-mail addresses: [email protected] (M. Elkin), [email protected] (D. Peleg). 1Supported 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 |